Warning: Long answer incoming.
I would highly recommend doing Further Maths for any STEM course at any top university like Imperial. I am currently in year 13 doing Maths, Further Maths, Chemistry, Computer Science and EPQ predicted 5 A*s applying for Computing at Imperial so I think I would be a good person to answer your question. I have completed the pre-interview admissions test for Imperial, completed the interview (which is the hard part as only 1 in 16 applicants get the interview) and am now waiting for my verdict.
Further Maths is indeed a step up from normal maths. I initially didn't do Further Maths because I was unsure about the field I wanted to enter (maybe medicine at the time? If you do Further Maths and apply for medicine, it's looked down upon and isn't really considered a fourth A level) so I did Maths, Biology, Chemistry and Computer Science. For the first two or three months in year 12, I was feeling extremely bored with just normal A level Maths with regards to the pace that the lessons were going at and the lack of difficulty with the work. I just felt as though I wasn't being challenged in any way whatsoever and I liked Maths for the challenge. During our GCSE year, we finished the course about 8 months early and then our teacher did some fun problem solving things with us that was extremely challenging and above our GCSE standard.
To give you an idea, we went down the route of why integration exists and how and why it works, which is stuff that isn't covered at the GCSE Maths level (our school doesn't give us the option of doing GCSE Further Maths otherwise I would have picked it up without a doubt).
Following my work experience, I realised that Computer Science was what I wanted to do and I was extremely grateful for what I had done in the previous years with regards to academic competitions as it meant that I didn't have to do as much wider reading as completely starting from scratch. Because I was unsure of what I wanted to do with myself, I took part in as many things I could do so that when I finally made the decision, I didn't have to start grinding stuff to do with regards to super curricular activities. In November/December of Year 12, I made the switch from Biology to Further Maths.
Now, Further Maths is indeed a step up from A level Maths - it's intended to be- but when we came round to covering the first year of Further Maths, I didn't find it that hard to be honest (other than maybe one or two small topics). I really enjoyed the challenge and it improved my own understanding of Maths altogether. There are some areas which are completely brand new (like complex numbers, Argand diagrams and matrices) and some bits (like the stuff on roots of polynomials) which built on concepts we learnt at GCSE and A level Maths.
The question you have to ask yourself is this: How much do I like Maths? I LOVED Maths but not to the extent that I wanted to pursue a degree in Maths. Half of my lessons in a week are taken by Maths and it means that you have to dedicate a lot more of your time to Maths as the pace at which content is covered is extremely fast.
We covered the entirety of A level maths in 8 months after taking Further Maths (you will find that some schools do the first year of A level Maths with the first year of A level Further Maths in year 12 and then the second year of A level Maths and the second year of A level Further Maths in year 13 but my school does year 1 and year 2 of A level Maths in year 12 and then year 1 and year 2 of Further Maths in year 13) before starting the Further Maths content so when we did our year 12 mocks for our predicted grades, we basically sat the whole A level Maths in our Further Maths mock (it wasn't our actual A level Maths exam but it was the same content, if not harder).
Also with regards to the largest drop out rate, your teachers are waffling - A level Computer Science has the highest drop out rate, not just in my school but in the country (I believe it is 10%). To take Further Maths, you need at least a grade 8 in GCSE Maths at my school so it is already limiting the demographic that Further Maths is available to. For Computing, all you need is a grade 5 in GCSE Maths or a grade 6 in an ICT related subject at my school.
For Computing or JMC (Joint Maths and Computing) courses at Imperial, I would say 1000000% take Further Maths because in addition to your A level requirements, you get something called a STEP condition, which you might have heard of but on the chance that you haven't, here's an explanation. The STEP is an admissions test organised by the University of Cambridge for their Maths applicants and a bunch of other top universities like Imperial, Warwick and UCL require you take it for some of their courses and you take in the summer of year 13. It stands for Sixth Term Examination Paper (because the 6th term in your A level year is the summer term). At Imperial, they ask you to take the STEP for any Maths and/or Computing course and standard offer is grade 2 in STEP II.
To give you a rough idea of what is tested in each of these papers, here's the breakdown:
STEP I - AS level Maths and A level Maths but applied to a completely new level such that you need to be on top of your game to get the questions right. The questions are hard and I mean HARD!! This is the first level of the STEP but this has been discontinued in 2021 due to loads of applicants performing well on it and the paper no longer being a successful differentiator of candidates.
STEP II - A level Maths and AS level Further Maths but again, applied to a completely new level and extremely tricky. A lot harder than STEP I.
STEP III - The stuff that nightmares are made of. AS level Further Maths and A level Further Maths but applied to the level of undergraduate Maths at university. The hardest admissions test for STEM courses in the UK. A lot lot lot lot lot lot harder than STEP II.
So without having done Further Maths at A level, you have no chance of succeeding in the STEP unless you do LOADS of work on it outside of school, get a tutor to teach you the Further Maths content and then get a tutor to prepare you for the STEP.
In addition to this, you would be at a HUGE disadvantage if you didn't take Further Maths because there are hundreds if not thousands of applicants who are doing it, not to mention the international applicant crowd whose level of Maths is simply insane. I was fortunate enough to attend the Future Computing Taster Course organised by Imperial and I'm not joking when I say this but: there were over 250 people who attended the taster course and I'd say over 200 of them do the following combination: Maths, Further Maths, Computer Science, Chemistry and/or Physics. You will be putting yourself at a severe disadvantage as the acceptance rate for the course is 1 in 20 and if you don't do Further Maths, you will not stand out to the admissions tutors.
If your school doesn't offer Further Maths (which I'm assuming is not the case otherwise you wouldn't be asking this question), you can let the admissions tutors know in your UCAS application that unfortunately you didn't have the opportunity to complete formal Further Maths qualification but here you have the opportunity to take the initiative and say that you took the initiative and taught yourself the course. If this is the case, I'd recommend doing Further Statistics 1 and Decision Mathematics as the modules you teach yourself. Stats will be useful in a Computing context as you will be dealing with a lot of probability calculations, particularly in the context of machine learning and Decision maths is basically A level Computing without the project and there are a bunch of other algorithms covered in that.
To conclude, if your school offers Further Maths, take it because there will be so many other applicants who do Further Maths and you will be putting yourself at a disadvantage by not doing it. In addition to this, your offer from Imperial comes with a STEP requirement, which can be from either of the STEP papers (STEP II or STEP III) at any grade but the grade is dependent on how much they want you.
If your school doesn't offer Further Maths, try to learn it as much as you can in your free time or speak to your Maths teachers to see whether it would be possible for them to coach you. Show your passion and I don't see why they wouldn't give up their time to teach you the course.
Good luck! And wish me luck for my offer.