which is best -to register a cattery name with TICA or CFA?

Ask your mentor and/or your cats breeder(s). Which association do your cats or cats from these lines do better in? I know with some breeds there are slight differences in the standards or the interpretation of the standards so that some cats/ lines do better in CFA and others in TICA.

Or if that is not an issue then look at the show calendars to see where you can show. In some areas there are more shows of one association than the other. (And as your mentor probably told you, you should get that 5 month old kitten out to some shows in the next 3 months to get her some show experience in the kitten classes and give yourself a chance to meet some more people. )

Or why not register your cattery name in both associations so you can register the kittens in both with your prefix? If you think you'll be able to breed show quality kittens and sell to show homes, you may have buyers who want to show in each and this could make it a bit easier for them.

Also you might as well do it now rather than take the chance that you register your cattery in TICA but not CFA and then in 2 years you decide that you want to show in CFA but by then someone else has taken the name. It would cost you $125 for both. For TICA it's a 1 time fee of $50, for CFA it iis $75 but i think you have to pay $25 renewal every 5 yrs.

Depends on which registry you which to show with. TICA has far more shows that CFA. And is this a show quality Sphynx from Champion, Grand Champion lines? If not, it wouldn't do well at shows.

Why do you want to register a cattery name? You don't just start breeding cats. You start off showing an alter and then spending several years learning about the breed. You mentor with a breeder and once they trust that you want to breed for the right reasons - for the love of and to better the breed - then an only then would you be allowed to purchase an unaltered, show quality Sphynx. And that will end up costing you at LEAST $2K.