Yes, it's quite possible to train to be a JAA FI(R) while working full time. I and many others have done it. I wouldn't like to say that it's particularly easy though and it does at times feel like a test of commitment.
In order to be paid you need to have a CPL to attach your FI(R) rating to. It's theoretically possible to instruct without a CPL but since you need the ground exams anyway there's very few who would bother to go that far and then not get the CPL.
Personally, I found the study for the exams to be the most testing part. These are a real test of commitment and most of us wonder about the point of a lot of the study - got to do it all the same though.
Having got the ground exams, the CPL flying training not all that bad but you might have to search a bit to find someone who'll instruct you for CPL level at weekends.
FI training is good fun and more relaxed than the CPL training but, again, it can be a challenge finding someone to do your training at the weekends. For both CPL and FI I did the majority of the training at weekends and then took a week off work to get ready for each test.
So..., it can be done..., but (nearly) full time work won't make it easy. It's not very cheap either. It'll take many years of very low paid part time instructing to re-coupe the original investment.
I don't know anything about the FAA route.