I used a 6 gallon glass carboy for my kit wine secondary, used a larger (maybe 8 gallon) fermentor bucket specifically for wine kits for the primary. I think you will be fine with the 6 gallon, but I guess it would depend on the kit...
Oh, and no, the fermentation is slow and smelly but not vigorous. You might get by with the 6.5 gallon container you have for primary and the 6 gallon BB for secondary. I didn't leave a large volume behind when I racked from primary to secondary, and there's air space at first, to leave room for degassing. After degassing you top up to minimize air space at top of carboy (and you "clarify" the wine for a couple of weeks, with airlock).
Good luck, don't delay, the wine kits are fun and easy (I mean cinchy simple gravy butter cakewalk easy, get SWMBO to help and it's fun to do together).
Primary - California Common -finally!
Secondary - nothing, back after year plus hiatus
Keg carbing & conditioning - nada
Drinking - store bought: Loose Cannon IPA, Wachusett IPA.
On deck: Trying to decide (beer); wine kit likely (red wine)