Should I get my elderly rabbit put down?

he s 10 and a half, has a slipped disc at the top of his neck which prevents him from cleaning his genitals, when I let him out he runs around for 5 minutes and then lies down for the rest of the day, he s drinking a heap of water, even after being on antibiotics for a UTI

I can t make the decision, I love him so dearly and he s my best little buddy, I ve had him since I was 11 , but I m house sitting for a woman for 5 weeks, and I was going to take him with me since he needs daily medical care but I think it would be so unfair on him to have to sit in a pet carrier all the time( they have rabbit eating dogs so letting the rabbit out is not an option. If I did take him I think it d just make his arthritis and sore hocks a lot worse

For people who say I m playing God if I put him down, I m pretty sure God would ve let him get eaten by a wild dog or cat if he was in the wild, before he got to the age of 10.5 years