Why not give it to a rescue then?
Lots of rescues have waiting lists before they'll take an animal in, which can be a problem if you're in a situation where you really can't have the animal for any longer.
I agree that lots of people would like this - the woman whose cat we rehomed wanted text updates for the first couple of months, and given how devastated she was to lose him it seemed the only kind thing to do - but if you don't want them just say something like 'So pleased to see s/he has settled in - please don't feel you have to keep updating me, s/he's yours now and I find it a bit upsetting.', OR just don't reply. I can't imagine they're planning to keep this up for the rest of the animal's life, in any case, so it will tail off, probably quite soon.