Can't keep Timing tape on balancer

I'd also suggest a dab of super glue on the ends.

When I do crankshaft torisonal measurements with laser optical tape on dampers I use a small dab of normal super glue after cleaning on the ends of the tape. I've never had one fly off when done that way.

It is possible that the lacquer thinner residue is eating the glue on the tape. Try using rubbing alcohol instead. Won't melt things like carb cleaner.