Pivoting into Management Consulting after 3 years of unrelated professional experience?

I am looking to start a career in Management Consulting but I have some questions. Any insight that anyone has would be appreciated.

I graduated from a UK uni with a 2.1 in Maths & Physics in 2016 and have been working as an economics & markets analyst in London since that time (i.e. analysing the latest economic data and market news and using it to make forecasts about future economic/price movements). I have also passed levels 1 & 2 of the CFA. I have finally realised, though, that this career is not for me and that I would like to begin a career in consulting in London.

My question is, for those of you working in London, do you know if I should be applying for graduate roles or if I should be aiming for a higher level position based on my 3 years professional experience (even though my experience is largely unrelated to consulting)?

One reason for my confusion is that I applied for graduate roles in audit last year and was offered a job at KPMG, but I rejected the offer for two reasons:
1. I changed my mind about going into audit,
but more importantly...
2. I felt much more experienced than the other candidates (i.e. undergrads) and felt I could aim higher.

My other worry is that plenty of consulting companies prohibit you from applying for multiple roles for 6 - 24 months at a time. If I aim too high the first time I apply, and get rejected, then I will have to wait another 6 months or more before I can apply again but at the right level.

So does my experience entail me to apply for a role slightly above graduate? Or am I expecting too much from myself?

Many thanks...