If you pitch your yeast first, and put it in your keezer for 2 weeks at 35F that should be ok.
Yeast, even the finely selected brewing yeast, has a wider temperature range than we think. It won't brew correctly outside the brewing range, but it will live (like all of us living in our 65F to 75F climate controlled buildings, but we can function in 95F or in 35F)
At 35F, you'd slow the bacteria sufficiently that your pitched yeast should out compete it especially as it comes up to temp. I'd not put it in the keezer without putting the yeast in first, even if I had to pitch again to get fermenting going later.
As to the other ideas, you mentioned having 2 in the fridge and able to get one more? Can you remove one of the others? Temp control is important, but it is also variable when important. If your current occupants are 95% done, I'd be less concerned with them in the swamp cooler (or the 35F keezer) than a fresh batch of wort in those conditions.