My DDs do competitive gymnastics, and from my experience I'd say he would be unlikely to make it as an Olympic gymnast from the small club, but I'd also question whether that is something you really want? To be a top level gymnast, you pretty much need to sacrifice your childhood, and there's no guarantees and very little future in gymnastics. My youngest daughter (also 6 1/2) does competitive gymnastics, and she trains 7.5 hours a week. One of her friends has just moved to a very serious gym club, and at age 6, she will be doing 17 hours a week (this will increase as she gets older). She has been identified as having potential to make it (she's now going to one of the Olympians clubs) and fair play to her, but I'm not sure I'd make the same choices for my child, given the sacrifices that comes with it.
My eldest (9) trains 18 hours a week (split over 5 days, including Sat and Sunday) term time, and over the summer holidays its around 23 hours. Already, my daughter has to sacrifice so much - she misses birthday parties, sleep overs, play dates so much because she is always at the gym. But this is her choice, and she loves it. Yet, even on 18 hours a week at age 9, she is never going to 'make it' as a gymnast. Sure she can do some pretty good tricks - tuckbacks, straightbacks and the like and she competes at a regional level, and will hopefully take her national grades next year. But to even stand a chance at international level, she would have to do far more hours, miss time off school etc.
However, they don't need to be an Olympic gymnast to have a great time doing gymnastics. My DD trains, has a close knit circle of friends in her gym friends, enjoys competitions and learning new tricks. There's far more to gymnastics than being that elusive Olympian, which so few children achieve (and believe me, there are thousands upon thousands of gymnasts out there!)
Being honest, if the big club says he is unlikely to make it - then he probably won't make it as an Olympian, most clubs know exactly what it is they are looking for. And even out of those they think are going to make it, few will. But, get him to enjoy gymnastics for what it is (and it is a fantastic sport) and he will probably still be able to compete and enjoy himself at the smaller club. The Bigger club would probably offer more opportunities, but as that door is closed, then enjoy the smaller club. And if he gets on brilliantly, then in a few years maybe look at another bigger club if he shows promise and perhaps move him then. Different clubs do look for different things, and he may well make excellent progress and you can review things later.