It shouldnt fail IMMEDIATELY with a low input voltage-- low input
voltage is going to lead to higher than usual currents, and if the
design is any good... well, maybe that's the problem.

Also a TV is not a happy load-- lots of peak current draw at the top
of each cycle.

Probably should shop for a higher-quality inverter, say one with a
one-year guarantee.

Harbor Freight stuff is cheap, which often translates to "marginal
quality".