Probably not. Tavern Brawler requires you to "hit the creature" with a weapon attack. While the Disarm rule involves making an attack, it's not clear that that attack involves hitting a creature. Normally, hitting a creature requires beating its AC:

When you make an attack, your attack roll determines whether the attack hits or misses. To make an attack roll, roll a d20 and add the appropriate modifiers. If the total of the roll plus modifiers equals or exceeds the target’s Armor Class (AC), the attack hits. The AC of a character is determined at character creation, whereas the AC of a monster is in its stat block.

Since the attack you make to Disarm isn't against a creature's AC, it probably doesn't involve hitting a creature, and therefore can't trigger Tavern Brawler.

You can pick it up.

The question you quote is correct that your enemy can just pick the item back up, but they have to do it on their turn. You can pick it up on the same turn you do the disarm, though (PHB 190):

You can also interact with one object or feature of the environment for free, during either your move or your action.

In the table on the same page, labeled "Interacting with Objects Around You," one of the options is

pick up a dropped axe

Your disarm is part of your action, so you can pick it up alongside this action. Now that you've stolen the disarmed item, the enemy has to disarm you to get it back.

You can stash it.

It's a bit ambiguous whether or not you can stow the item in the same round without expending an action on it. The same page in the PHB states,

If you want to interact with a second object, you need to use your action.

Since picking up the item and stowing it is interacting with only one item, one could easily argue that the entire thing falls under a single item interaction. If that is the case, then your enemy would have to do something else to get the item back, as you're no longer holding it. Alternatively, since it's two different actions you're taking with the same item, a DM could say that it costs an action. It's ultimately up to the DM anyway:

The DM might require you to use an action for any of these activities when it needs special care or when it presents an unusual obstacle.

Personally, I would rule differently depending on the item. If it was a small orb or something, I would let a character put it away without a second item interaction, but if it was a warhammer, I would require an action.

Your teammates can damage it.

If you have a teammate that goes before them, they could also pick up the item or attack it or something before the enemy can retrieve it.

For example, Fire Storm states,

The fire damages objects in the area and ignites flammable objects that aren't being worn or carried.

and therefore would damage or destroy the object once it's dropped. Similar spells like Fireball have similar wording. Indeed, if someone were to cast Fire Storm on the square in which your enemy was standing, the disarmed object would catch on fire as soon as it is no longer carried.

For completeness, the PHB does allow targeting objects with attack rolls. On pg. 194, in the "Make an Attack" section, the first option is:

1. Choose a target. Pick a target within your attack’s range: a creature, an object, or a location.