Harmonic balancer comes off

Also, when we dry fit it and it's pushed on all the way, it wiggles left a bit . (Wiggles as in turns maybe 1/20th of an inch, moving in a way a steering wheel does). Is that a problem?

You actually were able to slide the balancer all the way on? No this is not right, the balancer is a interference fit. I would start with another balancer, if it slides on, you have crank issues.