
This is interesting because I'm having a similar dilemma. My 6 month old Japanese Chin is on JWB puppy pouches and Royal Canin mini junior dry. The JWB says to switch to adult at 6/7 months, and the RC says at 9 months. I think I'm going to keep mine on both puppy formulas until 9 months and then switch over to adult.
A bichon frise is also a toy breed, and I would maybe keep them on puppy food for another couple of months - toys are usually fully grown at about 9 months. And yes, I believe the JWB junior dry is for larger breeds - the kibble size might be too big for your bichon.
I think the same brand of puppy/adult would be of similar content (just with different amounts of certain nutrients) and wouldn't upset the dog's stomach if they are used to eating that brand.