I think it's possible.
Technically you're supposed to race for one team, just to be precise.
I think it's sort of "done", especially if a rider has graduated to a higher category. I think, for example, that you'll be a 2 this year, and that's a totally different situation than being a 3 and totally different from being a 4. There's a local guy who races for Jelly Belly now. When he comes back home he's still friendly with his old team. I don't know if he actively helps them but it's one of those things where he can race with friends if he wants, meaning in a team work kind of way.
I have Carpe Diem Racing (it's technically a USAC club) and race with another team (Expo Wheelmen). I basically never have a chance to wear a CDR type kit (there isn't even a current iteration, post-2007 anyway). CDR is the logistical/legal vehicle for promoting Bethel. I've kept it alive because I have an attachment to the name (since I suggested it back in 1988 when it wasn't my club). I raced "on my own" for many years before joining a different local team, primarily for the race day camaraderie and some teamwork. All my CDR teammates had quit racing for various reasons, and for maybe 5-6 years they only showed up at Bethel anyway.
However I've kept CDR and left membership open to friends and such. 10 years ago there were still unattached fees and a rider might have to pay $10 or more to enter a race. Riders and former employees (of my shop) would list CDR as their team and they'd wear the kits from back in the day (1995-2007). Logistically there's not much to do - renew the club membership. I have no kits but that, along with sponsorship stuff, are things you could pass on to other members.
With your club I don't know how much is involved in doing the sponsorship and stuff. Kits, unless they change, are simple re-orders.
If you join another team I'd do it as "just a member" so that you don't have a bunch of conflicts of interest. At Expo I am intentionally just a member, as I was with all the teams since I joined a non-CDR team. With CDR I was super involved in the club, dealing with kits, membership, group rides, etc.
Finally there's the option of recruiting riders to ride with you. As a 2 though I think joining an established team might be better.