Do independent UCAS applications whilst in study look dodgy ?

You cannot avoid the predicted grades by applying outside your school.
Someone will need to write your reference and provide predicted grades, and if that isnt your school, it rings the biggest alarm bells possible to any University.
Stop being petulant. You have three good grades and more than enough for a Cambridge application. No Uni requires 4 A levels, so they will just ignore the C grade. You could just drop the C grade subject now and focus on the 3 grades that matter.

I asked my high school engineering teacher, and they said they would be more than happy to write me a reference if needed. The issue is since I'm applying for engineering, FM is pretty important so I think they would take the grade into consideration. I couldn't drop it either as they place quite an emphasis on doing it

Your Engineering teacher is being very naïve. Your school would not allow any staff member to go off-piste like this - and I suspect your teacher thinks its just like writing a job reference and hasn't a clue about how UCAS applications are actually organised at school level. This sounds like a recipe to land both of you in hot water and create a great deal of trouble/annoyance that isnt going to endear you to anyone.
If you want to 'defy' your FM teacher, then don't apply to Uni this year.
Work your socks off, get 4 A* grades and then apply with achieved grades in a gap year.

Thanks for your advice, for clarification my engineering teacher was from my high school and I am now at a different collage, not within the high school sixth form. So what would my best option be
In either case would it be worthwhile to apply this year and next, or will it hurt my chances of getting in next year too

Please explain how your GCSE teacher is going to be able to predict your A level grades.
And do they actually realise this is part of the process?

I thought predicted grades were not asked for if an application is done independently

Do independent UCAS applications whilst in study look dodgy ? I am in sixth form, and don't agree with one of my predicted grades (A*AAC), and as I wanted to apply for top institutions (Oxbridge), I am not too sure what to do. The exam that I got the C in (FM) had a really strict mark scheme, and I was 1 mark below a B. In year 12, we only did AS core pure, and in Year 13, we're doing Core Pure 2, and the 3 additional modules, but our UCAS needs to be submitted before our next exam which means there won't be time for it to be changed. So my question is what would be more beneficial in my scenario, applying individually, or would this look worse ?