A lot of Thai people are Buddhist.
Buddhists, on the whole, do not believe in taking life.
to describe their attitude as, "Their attitude is appalling." is not really appropriate.
To take a life (even of an insect, let alone a mammal), is seen as something against nature and against the Buddha.
If your relative wants to end a life, then take a hammer and murder the poor thing, or wring its neck.
Or maybe let nature take its course, as all the locals wish.
It is not really appropriate to ask someone else to take responsibility for your relative's sensibilities - especially if those people have an ethical objection to deliberately taking life. If he thinks the cat is better off dead, he should do it himself.