Logic Translation with Different Ground Reference
"...will this be low enough to trigger a 0v output for chip 1?"

Possibly not. It is well above the VIL for the TXS0102.

"Will this cause issues with the logic gate over time if its logic gate keeps seeing negative voltage for low?"

Probably. It is below the absolute minimum rating for the input of the TXS0102.

You could do some workarounds with voltage dividers if you don't mind drawing a bunch of extra current, slowing the traffic down, etc. But really, you should consider why you need an offset in grounds and eliminate it with some other solution. There's a reason this isn't a standard thing to do.