I missed out on head student. What do I do now??

For context, I’m in year 12 and I want to study econ at LSE ideally. I’m doing Maths, Physics, Economics and an EPQ and I’m doing very well in Maths and Econ, physics not so much but nothing to be concerned about. My school doesn’t predict A*s until our year 12 end of year mock so I’m currently on AAA. I interviewed for a prefect/head student role, I got prefect, missed out on head student. What should I do to compensate?? I want to make myself as competitive as possible. I’ve been considering possibly doing FM outside of school since October but I haven’t been able to find a good tutor here. Should I self teach it? I think I could but i’m not sure how my school would give me predicteds for it.

No Uni actually cares about 'prefect badges' - whatever your school tries to tell you.
What they want to see is evidence of your academic interest in the degree subject, and that you are moving beyond just the A level syllabus.
Suggested reading - good for all Unis - Economics Reading & Viewing Suggestions | Department of Economics and Undergraduate reading list - The University of Nottingham

You really must make use of YouTube channels for Maths, Further Maths and Physics. Because there are so many tutors offering high quality content for free. Also, remember to read the examiner reports and 'scrutinise' the mark schemes of all the past exam papers you do.
If you're studying Edexcel A-Level Further Maths, then I recommend you do Further Mechanics 1 and Further Decision 1 as they will be easiest to 'maximise' your marks. Or you could opt for Further Mechanics 1 and Further Mechanics 2. Because you're studying Physics too. More importantly, the textbooks for Pearson Edexcel include model answers from the examiners online with a link printed inside the textbook. So, you should have no excuses for achieving an A grade or more likely, an A* in A-Level Further Maths. 🙂

Thank you so much, I was a bit lost before on how people managed to self-teach it. I just have one question, for people who self-study FM how do they get their predicteds from their school? Do they normally just get given tests every few weeks so the teachers can gauge their level?

No Uni actually cares about 'prefect badges' - whatever your school tries to tell you.
What they want to see is evidence of your academic interest in the degree subject, and that you are moving beyond just the A level syllabus.
Suggested reading - good for all Unis - Economics Reading & Viewing Suggestions | Department of Economics and Undergraduate reading list - The University of Nottingham

Law - Cambridge reading list - good for all Unis - super-curricular_suggestions_2.pdf
BBC podcasts - BBC Radio 4 - The Law Show - Available now
Anthropology -Suggested Readings | UCL Anthropology - UCL – University College London
BBC podcasts - BBC Sounds - From Savage to Self - Available Episodes and Arts & Ideas - How anthropology helps us understand the world - BBC Sounds