You have TWO issues here.
1. Yes you can use your own a/c IF it is publci transport CofA. If private cat or PFA (or whatever the equivilent is these days) then nope you cant.
2. You MUST be supervised by an FI, AND work under and RF that has registered the FIs who supervise you as supervising FIs. (SRG1111 form required to the CAA).
The whole point of FI(R) is that you cannot just set-up your own school without first obtaining the required experience. And believe me...when I look at myself now compared with my FI(R) days....wow...I thought I knew a fair bit...but crumbs..how much did I not know - and Im not talking about flying per se...but more about teaching.