Assume I'm a fighter, Sword and Board.
If the shield is in main hand as an improvised weapon, doing 1d4+STR, and my longsword is in off hand doing 1d8+STR. I have Two-Weapon Fighting and Dual Wielding.
Would this allow me to shield bash with my shield twice with extra attack (e.g. shove or do 1d4 damage), and make an off hand full attack with the longsword as my bonus action, and meet the requirements for the +1 AC from Dual Wielder:
You master fighting with two weapons, gaining the following benefits: You gain a +1 bonus to AC while you are wielding a separate melee weapon in each hand.
You can use two-weapon fighting even when the one handed melee weapons you are wielding aren’t light.