OP leave a note on the door
" Please call Catsos at no 3 on xxxxxxxxxxx. My cat got into your house at the weekend but your kind builder, Mr Lovely Irishman, came over to get him out."
If you see them or the builder arriving go over and explain/remove food iitems/ offer to clean rug professionally and repair other damage.
They seem to have been friendly when you have chatted before. Show them this thread and they will realise how you were trying every avenue and that there are dafter people around (who would make much worse neighbours).
I'm sure your lovely cat will be fine - last time we moved house our obstinate cat refused to use a litter tray for almost 3 days! And we once seriously considered breaking into a neighbour's (unheated) conservatory, in the middle of winter, to 'rescue ' a hamster.