Welcome! You have the same concerns as many of us. A lot will depend on your DS's teacher's attitude when he starts school; I suppose at this point you've decided on a school, but you should definitely be open to looking at others if this one doesn't do sensible things with your DS.
You can also encourage things needing perseverance and good habits outside school - many people like instrument lessons for this reason, or gym, dance, swimming... whatever seems likely to be something your DS enjoys but doesn't effortlessly excel at!
Unless your pediatrician knows something about the particular school your DS is going to, it would be quite surprising if the advice that they may want to advance him to 1st is accurate; most people seem to find schools reluctant to do grade skips, and for a school to suggest it is almost (but not quite) unheard of.
Testing is not very likely to help IMO; you could consider it if the school isn't helpful and you get the impression that it's because they don't believe he's exceptional, but I'd wait till then, personally. It wouldn't do any harm to investigate suitable testers near you and find out what they cost and whether they have waiting lists, mind, so that if you do want to do it later you can move fast. Testing at 6 or 7 tends to be more reliable than testing at 5 (e.g. it's less dependent on rapport between your child and the tester, once the child is old enough to understand going along with something even if it isn't fun) so if you can leave it a bit, you may save yourself some angst as the results are more likely to be consistent with what you see.
