In the new wing of DFW Terminal D by gate D1 there is a stand (on the left as you walk towards D1) that appears to sell only full sized cakes. I didn't get a good look at it, but it was there. Not helpful for OP (unless one of the guests flies in from DFW).

But I second the Admirals Club suggestion, definitely renting the conference room for an hour but having one of the guests get there to decorate it (simple, a few streamers that say HAPPY BIRTHDAY, some festive paper, napkins, and plates) -- ideally one of you knows someone at the club there (I think John the bartender would think it to be a fun idea, but a front desk agent would be even better) to approve the decorations. 50-50 a couple staff even join in singing happy birthday. The BOS staff are pretty great. (Odds of all this working are much lower if you don't rent the room... you're not even supposed to have outside food.)

If you're bent on doing it on board, definitely do securely packed cupcakes and wait until the FA is un-busy (i.e. NOT during boarding) to mention that you brought cupcakes for a birthday celebration and would it be okay if a few of you sang happy birthday, etc. Obviously bring enough for 10 pax + at least 3 FAs, individually boxed with a tamper-evident sticker à la Georgetown Cupcake, Sprinkles.

I get the idea, but if I were paying for the flight and a group of folks started getting up and having a party I would roll my eyes and wonder how bad things would get next. (It sounds like you're planning on cake and that's it, but think what the other pax would think if they didn't know any different.)

La Premiere is too small for you with only 4 seats, but perhaps try Emirates A380, you could do it in the upper deck bar. Maybe for the next birthday.