What happens if you give a backline unit Tank?

This is a well-defined interaction under Rule 460.2.c.7. When a unit has both Tank ("assign damage to me first") and Backline ("assign damage to me last"), those are exclusionary demands. The assigning player chooses which one to apply.

So the opponent assigning damage gets to decide: either treat it as Tank (damage it first) or treat it as Backline (damage it last). They cannot split the difference — the unit must be assigned damage entirely within the priority slot they choose for it.

The FAQ on Caitlyn (who naturally has what is now Backline) being given Tank confirms this — the opponent can bypass either keyword's restriction by choosing the other.