Digging up an old tweet regarding interactions of multiple feats and features.

Say a character has obtained the feats: "Dual Wielder" and "Tavern Brawler"  alongside the "Two-Weapon fighting" fighting style and goes with a sword and board approach to combat.
Action: Atk, BA: improvised weapon attack with the shield for 1d4 + STR MOD damage

The easy part:
The above can be done since "Dual Wielder" allows non-light weapons to be used in conjunction with the Two-Weapon fighting feature using the shield as an improvised weapon on the off-hand attack (anything can be used as an improvised weapon).

There's supporting literature to back up the claim that the character gets a +1 to AC in addition to that when wielding a shield... with conditions:
A line in the "Dual Wielder" feat mentions that a character gets a "+1 bonus to AC while you are wielding a separate melee weapon in each hand." Does a shield acting as an improvised weapon meet the condition for the +1 AC part of the Dual Wielder feat? 
JC chimed in on Sage Advice saying "An improvised weapon is, indeed, a weapon, but only the moment it's used as such. A chair/shield/etc isn't a weapon otherwise."
This means that an improvised weapon counts as weapon so the character gets the +1 to AC the moment the shield is used as an improvised weapon.
JC's tweet then trails off into the ether without further clarification on what "only the moment it's used as such" actually means mechanically. Does the character gets a constant +1 to AC? Does it only give the character +1 to AC only on attacking with the shield? Is the condition checked at the start, end, during the turn / round? Who knows.

The Hard Part: 
If the character then takes the "Dueling" fighting style then things gets messy. Normal sword and board setup allows the additional +2 to dmg since the character is holding only "1" weapon. When an improvised weapon comes into play, the character is now holding 2 weapons - the conditions for the Dueling fighting style is no longer met.... however when the previous JC sage advice tweet is thrown into the mix the bonus +2 to damage provided by the dueling fighting style kicks in whenever the "improvised weapon" isn't used as such.

Confused? I sure am. Thoughts?

It comes down to DM call. Personally, I'd rule that a shield can't be used as an improvised weapon purely on the basis that there's already a different feat that covers shield bashing, plus it's just a bit too much cheese to attempt to squeeze all of that into one item.

There comes a point, as a DM, where you say, if I have to untangle that many things, it’s too complicated, and probably breaking something somewhere in there. Then you just say no.

I agree that a shield should qualify as an improvised weapon. However, I would rule that the sield only counts as an improvised weapon whilst it is swinging, not when you're defending. If you were to argue the case, I would say "fine, the shield is an improvised weapon this round, and not a shield.", and you would lose 1AC.

When you move onto duelling, it does indeed become complicated, and I would say that you have a choice:

1: Use the shield as a weapon
2: Use the shield as a shield.

You can't have both. It has the benefit that you can pick between offense and defence without dropping anything, but if you want to use your shield and get the defence, you need that shield bash feat instead!

I would rule that, since the PC is using the shield as an improvised weapon, and therefore it qualifies for two-weapon fighting purposes, that it therefore precludes interaction with the Dueling Fighting Style. It’s either/or, not both.