if you don't believe this person that this work was actually done,
then even paying you could be leaving yourself open to a claim against your insurance anyway, like a personal injury claim...
you could leave it go and see if they follow up on it or are chancing their arm with a mate in a garage.
if they follow it up:
I'd demand three comparable quotes from dealers on company paper, it is doubtful they have this. I'd expect each to give a breakdown of costs, parts, labour, vat etc.
I'd also want to see the repaired car and the old removed/damaged bumper to inspect level of damage and to see if there wasn't some preexisting damage you're not responsible for (know you're point of collision).
Check their invoice to see who did it, and if they're vat registered, and that it's a paid, ie an invoice and not a quote.
at some point you just have to be sick of this kind of thing.
obviously you have some duty to an injured party, but I don't see any problem with making things crystal clear when you're footing the bill. Getting a repair done without quotes, or without inspection to see preexisting damage is just not on.
After all this, for 1300E, you may want to put it through your insurance, as they might anyway, re a personal claim in a couple of years.