We have the same "birthday in Jan, too small house" situation. We gad dd's first birthday "celebration" at the local aquarium (she loves fish).
We said, this is what we are doing, this date, would love for you to join us. Some came, sone didnt - totally cool. Everyone paid for their own tix although i get a disc through work so most were discounted. We also had the issue of not really being able to afford much.
We then took everyone out to lunch. We actually found a place nearby that was not too spendy, casual enough for a bunch of crazy kids, and accommodating of multiple food allergies.
The cost of the tix combined with our insistence that gifts were not necessary did persuade some (mostly families with younger kids, money's tighter, etc) to follow the no gifts rule. They got a thank you card for "celebrating my birthday with me, so glad you could join us."
Other ppl said phbbbbbbbt to the no gifts but were sensitive to the fact that not everyone brought one. We were quietly handed gifts, quietly thanked them, and opened them at home. They got thank yous as well.
It was low stress, low key, and more about having fun together, less about the gift avalanche.