So long as he appears to be nice and healthy(decent color, curious and exploring, no clamped fins or heavy breathing or lethargy), then I'd say there wouldn't be any need to worry about moving him....there have been times when I've moved new boys around from tank to tank as I try to arrange everyone and fit them in after an impulse buy, or if I need to keep them in a smaller QT over night while I dig out an old setup for them.
IME, its never been too stressful of an event provided you acclimate properly, are sure to provide a good amount of plant cover in his new permanent home(make sure at least one or two of the plants, preferably silk or live, reach the, or close to the, top of the water!), make sure that if you have a filter its properly turned down/baffled to slow the water movement, and are sure your heater in the new tank is working and keeping things at a stable 78-82 degrees F. :) Just make sure to set everything up before hand, condition the water with a good conditioner like Prime, and let things sit for an hour or two just to make sure things are working and the heater is doing its job before you start the acclimation process(floating your betta in his cup in the new water for about 15-20 minutes, pouring out old water and adding in new every 5-10 minutes...you won't have to do this quite as carefully since hes already used to your water, thankfully, but its always a good idea to acclimate a betta to fresh water).
If you're really concerned, you can always leave the light on his new tank off for a day or so if he has one, or just make sure the room hes in is a little more dimly lit, but so long as he appears healthy then he should take to the new setup without a problem. :)