Unless you're kicking that guy off a building, I find it very hard to believe you're going to anywhere near outside of the blast radius... lol. Take a read up on grenades, see how big of aboom they deliver, ask yourself, What could go wrong here?
Cause let me tell you. A lot could go wrong with solving your problems with grenades. Everything blowing yourself up, to them pulling it back out in time, to someone else crossing the path, to some other type of essential collateral damage, to someone getting their hands on your stockpile of grenades.
Now, I don't think it's a problem at all to for your gunlugger to have infinite or nearly so grenades. But it's your job to make the world seem real. So ask him. Who supplies him with these grenades (do they supply anyone else?), we assume the gunlugger has a massive crate of these things around where does he keep them all? How many is he carrying on him at any given time? Hold him to that number. But really, it's not the person getting blow up that matters at all in this circumstance. What do the people who have to clean it up say? What do the people that loved that person say? Who are his friends? What decent humans are left in the area that finds his "act" stomache turning and grotesque? What do they do about it? Who tries to exclude the Gunlugger? Who tries to keep away from him? Who wont do business with him? Will he kill someone for disrespecting him when he is disrespecting everyone constantly? What will other people say to that.
Honestly, grenades sound like they're you best friend. On a miss, you have so many OHSHITS to deal with. You've got a nice sucker someone move to handling (not if he kills the dude, they're gone) the other things that go wrong. You've got a guy that is effecting everyone, permanently and loudly scarring the landscape. If your player is bringing all the PostApoc himself, you can try to bring some of the human back through your npcs as a way to make the world feel real and keep things interesting. You've got stockpiles of grenades somewhere, which is a big deal. I've hit my group with grenades a few times (and they're definitely ap), and every time it changes everything. If there are ways to get basically massive quantities of the stuff, the players might see them in circulation too.
So to reiterate.
1. Let your Gunlugger kill his foes with style
2. Don't just take his stuff away, there are variations of this that are more fun.
3. Learn about the Supply chain (generating badness?)
4. Learn about the Stockpile (a place where he's vulnerable)
5. Think about what could go wrong, push rolls when dealing with excessive explosives.
6. Think about the mess, make it matter.
7. Think about how people feel when they or their children see this, or come across it later.
8. Think about the radius of death, who/what might get caught up inside, revealed, or pushed out?
9. Think about those dead, or those that want to bury them.
10. Think about what this says about the people he associates with, make it matter.
11. Think about the people that might want to hire the gunlugger for being... So so so publicly violent.