Hi there,
I had exactly this problem when I was applying in year 12 as well so I can tell you how it went from my experience. I think it depends a lot on which course you're applying for and where you're applying to. I wanted to do ChemEng and I applied to unis ranging from Cambridge to Surrey. I wanted to do A-Level further maths but I didn't have the luxury due to the size of my school and them therefore not being able to accommodate it for me. I ended up, like you, having to study it independently. I made it clear to my uni's to which I was applying what the situation was and I think it's a positive thing from their perspective as it would mean you're going out of your way in bridging the gap between A-level and UG, showing how keen you are, despite how things are.
In my experience I would advise you not to make your life more complicated than it otherwise has to be, admissions is very much a game sometimes. Unless you're applying to a course which requires further maths I would be reluctant to sit an exam in it as it just adds more work to your plate - with that being said if you're looking to go into any numerate discipline you should absolutely learn the content as from my experience idk how I would have coped with my maths course otherwise last term. If you decide to not sit an exam there is no point adding it to your UCAS, but do let your uni's know your reading around the edges - it reflects well on you as an applicant.
Of course, I don't know the ins and outs of your story with further maths as it could be quite different to mine but that was the logic I had when I applied and even despite not sitting an exam in further maths I'm now starting my Lent Term in Cambridge doing ChemEng via NS so it seemed to work for me. Obviously whether or not you decide to sit an exam is a big decision so don't just take my word for it but talk it through with your parents, teachers etc... people around you that you trust. I hope this provides some clarity as I remember being in your same position and being rather perplexed, best of luck with your application.
P.S just to clarify if you ARE applying to a course that REQUIRES further maths, definitely try and do the exam and you should fully disclose it to them, unless you have a good reason (in which case see above)