Well, the problem is if you "just invert the phase of either of the
channels" and then listened back in mono, the channels would cancel
each other out and you would end up with a very thin sound. If you're
listening back to the track in the context of the whole mix, it will
probably sound like it just dissappeared!

If you can re-record it, that's the better alternative. If you can't,
then try any of the suggestions you've gotten here and make sure you go
back and forth between mono and stereo to see how the sound changes.
You should always do a stereo/mono comparison flipping back and forth
when recording any stereo track .

If you are going to re-record it, there's an easy way to make the mics
more in phase with each other - While positioning the mics, flip one
side out of phase, listen in mono, start with no EQ or effects on
either and get a good solid sound with your first mic. Then with the
second mic, find the position where the sound cancels the most or
sounds the thinnest. The closer you are micing, the more pronounced the
effect will be. Then ,flip the out of phase side back into phase and do
your mono/stereo comparison. This also works better if the mics are
closely matched.

This technique also works great when recording bass using a mic and a
direct box. Start with the direct sound and then, starting about an
inch away from the speaker, move the out of phase mic away from the
speaker until the sound almost completely cancels and then flip it back
into phase. This will get you the strongest signal. Since bass is
almost always in mono it can mean the difference between a solid or a
weak bass sound - and of course - a weak bass sound means a weak mix!