You forgot an option #3. Hone your pitch to management to get them to give you the time you need.

Honestly I'd probably look for a new job, but with that option off the table #1 runs the risk of you looking bad doing the bare minimum and having truly failing tests that pass the suite. #2 runs the risk of you doing a ton of work for free only to get a pat on the back and maybe a "good job" email.

What makes your management tick? Is it happy customers? Frame the discussion with "We had 47 customer impacting bugs that were missed by the test suite last month." Is it cost savings? "My team spent 312 man hours last quarter fiddling with the test framework when you could have been moving the product forward or fixing actual bugs".

Just because it's not "customer facing" doesn't mean it isn't "customer impacting".