[RPG] When firing a weapon with the artificer infusion “Returning Weapon” via catapult, does it come back to your hand after the spell’s conclusion

Repeating Shot causes the crossbow to ignore the Loading property. Therefore, so can we. The Ammunition property states that you need a free hand to load a single-handed weapon, but Repeating Shot makes clear that deciding to "manually load it" is optional. Thus, if you decide not to manually load it, there is nothing in the Ammunition property that would require you to have a free hand.

It looks like you're clear here. Enjoy your shield (or whatever).

A couple things limit you to 3 attacks

There are a few subtleties in the rules that you have missed that prevent you from getting a 4th attack. First of all, as explained in the rule you quoted, two-weapon fighting requires you to use your bonus action, and you only have one bonus action per turn, so you can only gain at most one additional attack with two-weapon fighting. Second, you also only have one Action per turn. Arcane Armament allows you to make a 2nd attack with that action, but you are still only taking the Attack action once per turn, so there is only one opportunity for two-weapon fighting to trigger.

So, to take you through the sequence of events that gives you 3 attacks, assuming you take the Dual Wielder feat: first, you take the Attack action. Because you are attacking with a magic weapon, you can make 2 attacks. Both of your weapons are magical, so it doesn't matter which weapons you attack with: you can attack twice with the same weapon, or once with each. After either one of these attacks, you can use two-weapon fighting to attack with the other weapon, using your bonus action in the process. Ultimately, this gives you 3 attacks, provided that at least one attack uses each weapon. And one of those attacks will not add your ability modifier to the damage. Generally, you would want to make your 2 attacks using the mace with Arcane Weapon cast on it so you have an additional chance to deal the elemental damage, and 1 attack with the other mace.

Dual Wielder is probably counter-productive for an artificer anyway

Unless you chose Archivist as your artificer specialization, you already have a way to deal damage as a bonus action that deals more damage than your 3rd mace attack would, by commanding your companion or turret to make an attack. So taking the Dual Wielder feat and fighting with 2 maces will most likely decrease your damage output. It also reduces your potential AC by 1, since you give up the option of a shield in order wield 2 weapons. Lastly, taking the feat means you miss out on an ability score improvement, which is a big deal for an artificer who needs high scores in both strength and intelligence, and probably also constitution if you're fighting in melee. (Exception for Battle Smith, which gets to use their intelligence modifier for magic weapon attacks.)