If you take the Attack action on your turn, you can use a bonus action to try to shove a creature within 5 feel of you with your shield.
It says nothing about actually attacking, just taking the Action. There's even further nuance when you look at when Bonus actions can be taken and how Attacks can be split between moves, etc.
Let me be clear... when i say something is a fine house rule, that is not meant with deraogatory intent. i do not see house rules as second class to RAW.
But, to me, even without clarity from official sources, it is not productive to twist and mangle RAW to shoehorn in extra edge cases rather than just house ruling it under what amounts to basically a "it is not strictly forbidden" kind of opening.
Consider that you are trying to invent within the rules a difference ebtween "take an attack action" and "making an attack" to allow other stuff to be done ***between those*** and my suspicion is that there are likely a broader problem with that kind of logic scope-wise than just this one aspect.
But first lets look at the attack action rules
"With this action, you make one melee or ranged attack. See the "Making an Attack" section for the rules that govern attacks.
Certain features, such as the Extra Attack feature of the fighter, allow you to make more than one attack with this action."
There is no difference in that rule between "taking an attack action" and "making an attack"... making an attack is what the action does. it requires a lot to try and read the Attack action and then see it as not meaning "make an attack" when you take the action.
As JC stated in part of his comments on the subject, the fact that movement between attacks is explicitly allowed does not equate to a universal allowance to insert any bonus action in the middle of the attack action or any action.
The text makes it clear that taking an attack action means making an attack they are not different things that happen at different times.
But let me ask you this - how far does the "take an action" vs "does what the action says" go?
I cast eldritch blast and get to make three beam attacks... can i move between them? Can i also cast a bonus action spell between the attacks, seeing one resolved, bonus action spell, then the others? I can take bonus actions any time right and if there is a presumtpion that that include "in the middle of another action" than well, how about then?
on a very broad scale, there are a lot of really odd or even paradoxical cases that slam all thru the rules is a general assessment of using bonus actions before the thing that earns the bonus action is imagined in the rules - by trying to create a difference between "taking the action" and "doing what the action does" is read into the rules.
It is IMO infinitely more direct, cleaner and more efficient to simply add a house rule that allows it in the case of this one feat than to take an axe to the "take an action" to cut it into "declare an action" and "do the stuff in the action."
its trying to fix a hangnail; with a chainsaw and for me - i have zero desire t make that kind of broad scope of a decision without reading thru the rulebooks for every other "when you take..." and "declare not same as..." etc... especially given the fact that extra turns and extra actions can be taken as well so if there is an opening between declare and act and effects can precede cause...