No, you can't use the bonus-action shove to attack in any other way.
As you noted: Per Jeremy Crawford's clarification on Twitter and subsequent 2019 update to the Sage Advice Compendium, the bonus-action shove from Shield Master can only be done after completing the entire Attack action.
The first bullet of the Shield Master feat (PHB, p. 170) says:
If you take the Attack action on your turn, you can use a bonus action to try to shove a creature within 5 feet of you with your shield.
The word "shove" refers to a specific thing in 5e, described here in the rules:
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.
The target must be no more than one size larger than you and must be within your reach. Instead of making an attack roll, you make a Strength (Athletics) check contested by the target's Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). You succeed automatically if the target is incapacitated. If you succeed, you either knock the target prone or push it 5 feet away from you.
The feat makes an exception regarding the timing of your shove, but everything else about it is the same. The mechanics for that shove are collectively described in the feat and in this section of the rules.
The rules on improvised weapons describe the mechanics of attacking with something that is not a weapon or otherwise designed to attack with. They don't describe the timing of such attacks.
The Attack action lets you make an attack (or, with the Extra Attack class feature, multiple attacks) using your action. The rules don't specify what you have to make this attack with, so you can make these attacks with any combination of weapons (normal or improvised) or unarmed strikes.
No rule allows you to substitute that regular bonus-action shove (granted by Shield Master) for any other sort of attack or action - including the use of your shield as an improvised weapon to attack (and do damage). In the absence of such a rule, you can't do so by RAW.
(As always, your DM could house-rule otherwise, but this would essentially be granting you a free attack every turn.)