[RPG] What type of action is sheathing a weapon and equipping another

Because it leaves your hand open you should go ahead and put it in your weapon slot anyways. You're going to want it enchanted as a weapon and not a shield anyways so you want it there.

Second, just because it's in your hands slot, does not mean it's consuming that slot. It's on your hand, but the text obviously allows you to wield your bastard sword two handed.

Finally, you can definitely use the scout power Dual Strike with your bastard sword two handed. You will get the damage bonus and also be able to attack with the gauntlet axe.

Important: If your DM for some reason does not allow you to attack with the gauntlet axe while using the bastard sword two handed, take your hand off the sword for the second attack. Changing the number of hands a weapon is using is a free action thus you should be able to easily take your hand off the sword, strike with the gauntlet and grip the sword again. (This should be unnecessary as the text is pretty clear here). If your DM still does not allow this, just move on and try a different strategy.

RAW? Not really; "Two-Weapon Fighting" is for melee weapons.

You provided the quote yourself, but I'll add the emphasis:

When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand.

Crossbow Expert vs. Your Feat

You've found the nearest equivalent to pistol-and-sword in RAW 5e. You've built some of it into your custom feat, but you left out the part that would allow a pistol-and-sword style.

When you use the Attack action and attack with a one-handed weapon, you can use a bonus action to attack with a hand crossbow (pistol) you are holding.

As a result, you have allowed the less-realistic of the two options. Loading a black powder weapon is a long, complicated, multi-stage process; you have to load the powder, pack the ball, pack the wad, then load the primer. Typically, black powder pistoleros carry more than one gun.

How to solve your problem simply

1. Apply the bonus-action-shot rule from Crossbow Expert.

2. Have your pistolerogue carry more than one pistol.

   Instead of messing with Loading, adjust the custom feat to make drawing and holstering not require an action. Characters with larger numbers of attacks or who expect to maintain fire for multiple rounds will have to purchase multiple weapons. If pistols are suitably rare, that will serve as your balancing factor.

Experience base for this homebrew modification

I've done something similar for a pirate game in D&D 3.5, but the idea applies equally to D&D 5e.