Finn [name_m]Isaac[/name_m] is so handsome! I love [name_m]Finnick[/name_m] as well, but not with [name_m]Isaac[/name_m] as a middle.
Finn [name_m]Isaac[/name_m]! Such a sweet name.
Finn [name_m]Isaac[/name_m] has such a great ring to it!
I LOVE [name_u]Finn[/name_u] [name_m]Isaac[/name_m], so handsome.
I love the nn potential with [name_u]Barrett[/name_u] for [name_m]Bear[/name_m] and even [name_m]Rhett[/name_m], and prefer it with [name_m]Isaac[/name_m], but I like [name_u]Finn[/name_u] in general more than I like [name_u]Barrett[/name_u]. [name_u]Finn[/name_u] [name_m]Isaac[/name_m] is super handsome, too, though.
I wouldn’t be concerned with [name_u]Barry[/name_u], honestly–as dated as it sounds to you, it sounds just as dated to everyone else in your circles, I can almost assure you. If [name_u]Barrett[/name_u] is what you love, I would go for it. I think I’m leaning toward [name_u]Finn[/name_u] [name_m]Isaac[/name_m], myself, though, because I like [name_u]Finn[/name_u] better personally. 
Good luck!
I love [name_u]Barrett[/name_u], nicknamed [name_m]Bear[/name_m]. 
Finnick [name_m]Isaac[/name_m]
I have a [name_u]Finn[/name_u] and have gotten so many compliments on his name! I’ve yet to meet another, so even though it’s growing in popularity, I’m the same as you. I love his name: it’s short and spunky and fun and suits him so well. No regrets! I would go with [name_u]Finn[/name_u] [name_m]Isaac[/name_m] if I were you, it flows better without the similar endings.
As someone who has never heard of the name [name_u]Barrett[/name_u], I would probably mistake it for [name_m]Garrett[/name_m] if someone said it. Doesn’t mean you shouldn’t use it, but important to note that people might mishear your son’s name a lot and that can be annoying for a child.
Barrett is personally not my style, so I prefer [name_u]Finn[/name_u]. I don’t think [name_m]Finnick[/name_m] works due to what others have mentioned about the same sound at the end of first and middle. It trips over itself imo.
Barret [name_m]Issac[/name_m]