Distortion/saturation as an added effect is just not a good idea on live vocals. It's pretty much just asking for feedback. The best you could do is to turn it on when they're actually signing and back off in between. A noise gate
might be able to do that automagically, but likely there's so much bleed in the mic that you won't have good results. A push-to-talk switch is better, but even then it's often pretty hard to actually control. At an actual show, you might be able to get away with distortion on the FOH but not in the monitors. In general, though, it's best to just leave those things off* during live performance.
Now, when you say "reduced the gain of the offending mic", it leaves me wondering exactly what you mean there. Just turning down the gain control on your preamp/interface but then compensating for that at the fader or whatever doesn't help, because overall gain is still the same. Turning down the gain and compensating at the source
can help. That means getting the vocalist closer to the mic and singing louder/projecting better.
I've never messed with a feedback eliminator of any kind, but it really shouldn't be necessary. The rest of your ideas are pretty good, though.
Honestly, I think the EQ thing is going to be where you get the most mileage. This might sound counterintuitive, but try cutting out a lot of low end. The intelligibility in a vocal is in the upper mids, so a lot of times when we have a lot of low end eating up our headroom and muddying up the sound, we end up turning up the vocals more than they really need to be. Also, all directional mics are significantly less directional at lower frequencies, so you're catching more of what you're trying to point it away from - including the monitors that are causing the feedback. I consider the bottom of a vocal mic to be just reinforcing the drums and bass, and hack it out mercilessly.
You also don't need all the "air" up in the real treble region in most live situations. That's just cymbals and sibilance. Cut it out.
My default "face" track has ReaEQ with a high pass set at about 300Hz(sometimes higher) and a low pass somewhere up above like 8K where it's no longer competing with the guitars. I regularly reinforce extremely loud bands in extremely small rooms, and I have surprisingly few problems with feedback by applying these very basic principles. Like, I'll walk into a gig and they're having all kinds of trouble with feedback and still can't really make out the vox, I go over and turn a couple knobs and it's all fixed. It's not the superawesomest vocal sound ever, but it's working, and the show can go on.
Edit - Probably should have mentioned that eating the mic (to get the source level higher) increases bass in the vox via proximity effect, handling noise, AND the danger of plosive pops. The extreme high pass filter helps with all of that.
Another thing about getting closer to the mic is that sometimes the feedback actually comes from the monitors blasting at the vocalist's face and reflecting back into the mic. When you're right up on it, that pretty much can't happen.
Also, hopefully your vocalist knows not to try to cup the mic. That will make it less directional in the midrange frequencies where it's most likely to feedback already, and seems to create a nice little resonant chamber between hand and mouth which is what makes so sound so "cool", but also makes it howl like a bitch.
*Or embrace the feedback!