Are there any connecting flights to get you where you're going closer to schedule? I understand not wanting to kill a whole day, but a single connection may not set you back that much.

If not, I doubt you'll be able to get a refund. What you may get is a rebooking for another day, though. If you point out to the GA you barely have any time there if you get rebooked, ask for what you want (in terms of rebooking, not refunds). Do you want the return pushed out a day? That usually isn't a problem. Do you want to try this trip again next weekend instead? That might even be ok.

Don't think of it in terms of refunds and such. You basically have a one-time shot at a rebooking. Decide what you want ahead of time and ask for it. The GA may just do it.