As others have suggested, this might not even be much of a problem. Others have mentioned the extra time and spotlight this player is stealing from the game. My initial concerns were about 1) if this PC is doing these actions in combat then they are not pulling their weight in helping with the combat and 2) the realism of carrying 10 tons of gear.
For #1, some people consider weight restrictions a minor issue that they might be willing to drop. I have seen some games where they don't care, they loot everything, and PCs are currently carrying 15 swords, 10 daggers, 20 javelins, and 5 bows. When they get back to town they may say "I'll keep 3 swords and a bow in my personal treasure chest, keep 2 swords, 3 daggers, all the javelins and a bow on me, and sell the rest." And that is fine for some games.
If the weight issue is not something to be overlooked in this game, then instead of saying "If you keep that up I'll enforce weight restrictions," say "I get that we all want to keep everything, but I really don't feel like enforcing weight restrictions and keeping track of all the weights, so please at least keep it reasonable." And if that player pushes it too far after that, you can just arbitrarily say at some point (without actually keeping track of any weight, but just because it's past reasonable) "As you pick up that sword, you realize the weight of all your stuff is a bit too much. All that weight now makes you off balanced. If you don't put something down, you'll move at reduced speed and you might notice problems with your dexterity."
For #2: Why is it acceptable that this player is not helping in the combat, especially being a melee combat class? Are the battles too easy so it doesn't matter? If so, tweak the balance of the battles. Is the help needed, they barely survive, but this PC keeps doing these things anyway? Then the other PCs (the characters first, not the players) should yell at him. If that doesn't work, then you could have an out of character talk between the players on the grounds that this player is not helping with their role.
Player1: Orcsmash's eyes light up at the sight of the bandit leader's
bulky coin purse, and he tries to snatch it.
other players sigh and make comments
DM: (whatever the outcome is)
Player2: Magicdude is about a burst a vein and he screams at Orcsmash,
"Dude! What the ---- you doing!?! Get your head in the damned fight!
If I lose a hand because of you I swear you'll be the next target of
my uber strength curse, and that's if I don't lightning bolt your ---
And make sure you let the other player say what is necessary, even if it seems like too much to say in one round, as long as it's not a lengthy speech of course. You could insist at this point, out of character, that the other player at least play their character in a way that helps the party better, and insist on no more in-combat pick-pocketing at times when it could put the party in danger.
If this doesn't help, maybe they should follow through with their threats. And/or, maybe the characters decide they don't want that barbarian along anymore and would rather replace it with someone who pulls their weight fully. If it goes that far, the same player could even play the new PC that replaces the old one, and maybe it could even be essentially the same character, just with a different name and with the insistence that this "new" character does not neglect its combat duties.