it might be fairly difficult for a PSA Mint 9 to score a BGS Gem Mint 9.5…
some PSA 10s barely score a BGS 9.5.
as with submitting to BGS, you don’t need a membership with them to have cards graded.
you can print out submission forms and fill it out for crossover, you can set a minimum grade for which you wish to receive.
you can also fill out the forms online now… and BGS has a live chat representative who can assist you with the process and answer any of your inquiries.
if you’re looking to send just one card, I think it would be best to send it to BGS yourself without any middleman.
but yeah, best to contact a Beckett (BGS) representative and discuss with them before considering alternate options.
aside from that, good luck with the crossover! all the best!
Sending cards to beckett is very easy as @veryviciousvenusaur mentioned, but I’m not sure about crossgrading stuff with them, might aswell send in the card raw. I can only imagine them not being enthusiastic in handing over a BGS 9,5 to a PSA 9.