I play a new Stingray 5H running through a GK fusion 550 head and neo 410 cab. The foh and monitors are all JBL. Band is 3 piece and vocalist. I'm not much on hearing subtlety so I tend to run my head flat with maybe a tad bass boosted.
If I remember correctly, flat on a GK head is around the following (they seem to have a baked in mid-scoop):
Bass - 10 o'clock
Low Mid - 2 o'clock
High Mid - 2 o'clock
High - 10 o'clock
I have found that when playing with distorted guitars, mid-scoops do not work well.
Not sure if all GK's have the same baked in scoop, but on the MB200 it's 10, 2, 1, 11
Yep stage tone is a lot different than home tone. Have you tried raising your cab?
I'm not sure what the flat (or as close as practical) settings are on various GK heads, but, when I had my MB200, I gravitated toward something like the settings above when playing at home. Live? I'm not sure -- I just turned stuff until it sounded good. I do know that I never used the Contour switch on that head.
I can only specifically recall playing the Neo 410 once, and that was in a Guitar Center (though I wouldn't be surprised if I've encountered it elsewhere), so I'm not exactly experienced with it. At the time, it struck me as a little boomy down low and I remember that there was something about the mids I didn't like -- take that with a liberal sprinkling of salt, given the circumstances, but it might be that the head and the cab together have a "knobs at noon" voicing that's working against you. Depending on where you're playing (if it's a boomy stage or room to begin with), you could be fighting an uphill battle to hear yourself.
somebody on TB awhile ago suggested leaving your bass knob all the way down when you know you're going to be turning up kind of loud. Get the volume where you want it and then bring your bass knob up to taste. this may help remove the mud.
If you're having trouble hearing yourself, this is a good way to start. Get the mids where you want them -- you should be able to hear your pitches distinctly -- and then add bass to taste; if it starts to get blurry or boomy, back off. You might do some tweaking from there, but that should get you a sound you can at least live with. When you've got one tone that you know works, see if you can find another.

There's often more than one way to slot into a mix -- if nothing else, it'll be good practice familiarizing yourself with your EQ, which it sounds like you don't use all that much.
Try listening to the rest of the band without you playing -- how is the sound then? Does it sound like there's a bass player somewhere in the mix? Is it boomy and muddy without you? If so, there's some cleaning up to do with the band in general just to leave some sonic space for you (see earlier notes about boomy guitars and keys).
Have someone listen out in the house when you play. If you can, listen out in the house when someone plays your bass and your rig. Make a gig and/or rehearsal recording and listen to that.
"Cut bass and boost mids" is a good rule of thumb. It will help more often than not and I'd wager that it'll help in your situation. It's important to learn to hear what's going on in a given moment, though. I can think of a time or two when cutting the low mids and boosting the bass cleared up a (less-than-ideal) live mix I was playing in. More bass in your or your bandmates' monitor(s) can help, but it can also compete with your rig and clutter up the mix on stage. It depends on your rig (how well it disperses, how clearly the other players can hear it), the stage, the room, the mix... every situation is (potentially) different.
So, start with cutting the bass and boosting the mids -- don't be shy with the EQ -- and see how it works. Then try some of the other stuff and see how it goes. If your bandmates get impatient, remind them that this is to help the band's sound overall (that, and you shouldn't have to continue to play frustrated).
As far as equipment goes, I'd be curious how your head would do with a tighter, more mid-forward cab. 4x10s aren't so much my thing, but if there's an Ampeg SVT 410 HE (the sealed one, not the ported 410 HLF) that you can try, I'd be curious to hear how it works for you relative to the the GK Neo. (I spent a few days playing a GK 2000 RB through one some years back and
that combo would slice through anything.) A nice, vertical 2x12 (especially if it has a mid driver) or something similar might work well for you tonally as well as getting things up closer to your ear. A Gramma pad and/or an fDeck HPF3 could help with cleaning up the low end. See what you can do with what you have first, though.
Good luck!