There is case law on this point, if I remember correctly involving a car given to a garage for repair, the question being who is the 'owner' for the purposes of TMA? The judgment then examined the issue of 'keeper'.
And it is not the person who simply has day-to-day control, in that case the garage. There has to have been some degree of permanence as regards disposition.
IMO, the 'keeper' as regards PoFA does not operate as posited.
Just found it:
https://www.londontribunals.gov.uk/sites/default/files/keycases/francis-v-wandsworth.pdfIt might be convenient to consider that the 'keeper' is whoever has possession at any one time, but IMO as the judgment makes clear it is more complex.
Anyway OP, what may the 'hirer' as in PoFA do? They cannot relieve themselves of liability by naming you as the hirer. They could defend their position in other ways e.g. unless the creditor has sent them a Notice to Hirer(as per PoFA) accompanied by the necessary docs? Probably not, but unless they authorise you to represent them they might not bother to defend their position, it's simpler just to pass on your details* and see what happens.
But what might as opposed to may happen, who knows? Ignorance of PoFA abounds among PPCs and hire companies.