You can, but it is not ideal. The remaining channels on your mixer don't just lack an XLR connector, they also don't supply phantom power and they expect a line-level input. A simple XLR to jack adapter will not suffice.
To use a third microphone in this setup, you will have hook it up via a separate pre-amp to boost the signal up to line-level. The quality of the pre-amp greatly affects the sound, but if you plan to record speech, a cheap one will do just fine. It doesn't need to have a lot of features since you're still running the signal through the mixer, but if your mics require phantom power, do check whether the pre-amp can supply it.
In all likeliness, you'll end up spending more than if you had bought a more expensive mixer to begin with. This is why you'll often hear 'buy the best you can afford'. You're wasting money buying equipment you'll have to upgrade anyway.
Alternatively, you could simply plug the third mic into your Lexicon Alpha. It may be impractical to have the mix controls on separate devices, but this setup will work.
An external phantom power supply by itself definitely won't. It's not clear to me whether your mics even require phantom power, but without passing through a pre-amp, the signal into your mixer still wouldn't be line level. Depending on the wiring, you could get no signal at all, or a very weak one with the left or right channel inverted or missing.