Tinstaffl has it dead right. BFR and FAA medical and IPC if required. If you were to consider a type rating that is a different matter.
it is ESTA not VWP - but since you mention you are a Brit then that is easy enough and what you propose is entirely within ESTA. If you were to consider flying your twin to canada or mexico then you need a physical visa in your passport to re-enter the US, unless you have an entry stamp within the last 30 days ( if you are considering that , look at the info on the CBP website) .
It cannot be argued that you need a M1 visa. A school cannot undertake that process unless you are enrolled for more than 18 hrs a week. If you aren't, you can't. And for the record TSA don't care about your visa status as long as you are in the country legally.
What you may run into is the issue of the CBP officer who enters you into the country who doesn't understand the whole spectrum of the term 'flight training'. They understand the visa requirements for this about as well as the lurkers on here. For that reason, be honest and say that you are on vacation.. or even a flying vacation . but avoid the use of the word training, especially as you are not.