It all depends on how you present it to the customer, I always emphasise that they are paying for the work I am putting into their car not for the "FIX". In these nasty intermittent scenarios I always put a note on my invoice saying exactly what I found
"fault not evident at time of testing" etc.and that further work may be required if the problem reoccurs. Basically I pave the way for another bill if I am not confident that I have got to the bottom of the problem!
Who pays for the part? the customer of course, you make your tests as carefully as you can don't you? Sometimes though the only way to be sure is to substitute the part and you don't always have a known good secondhand one, so it has to be new. I often tell the customer that it is cheaper to fit a 30 quid part than spend an hour (£45)testing it, that pacifies them quite well.
I used to think it was about repairing cars, it's not, for me it's about feeding my kids. I just got fed up with fitting parts and providing labour for free to customers who plainly have more money than I do. Don't get me wrong, it's not that I don't care, but I do a good job and now I want paying for it!

I feel much better for getting that off my chest!
Shaun.