[RPG] Is this homebrew “Throwing Weapons Master” feat balanced

UA about feat design

First of all, I recommend taking a look at the Unearthed Arcana (D&D 5e playtest material) about feat design.

The main issues mentioned in the first section are not a problem for this feat, nor are those listed later on. Still, let's keep the UA in mind since it gives us insight into how the designers design feats.

Power comparison:

Let's take a look at the existing rules for throwing weapons without the Thrown property. The PHB states on page 148:

If a character uses a ranged weapon to make a melee attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage. An improvised thrown weapon has a normal range of 20 feet and a long range of 60 feet.

So basically, your homebrew feat currently allows someone who has it to bypass the damage reduction to 1d4, keeping the original damage instead. Therefore, the best you can get with this feat is 1d12 or 2d6 (both can be situationally better), with a range of 20/60.

That's not bad, but the most damaging ranged weapon, the Heavy Crossbow, deals 1d10 at a 100/400 range, and you don't end your turn without your weapon. Therefore, I believe the feat is clearly underpowered.

Furthermore, with the Warlock's Pact of the Blade feature, you have to use an Action to summon your weapon back. This is horribly inefficient, so I suggest that, instead of a feat, you design an Eldritch Invocation. I don't think you can really balance the feat as long as it takes you an action to summon back your weapon - and carrying around 20 greataxes around is not really viable, unless you're Steve from Minecraft who can apparently carry a bunch of Eiffel Towers in his trouser pockets. While you could certainly design a feat that allows warlocks to summon their pact weapon as a bonus action, feats are not meant to be so limited in scope. Therefore, an invocation is the only real choice, as I see it.

Eldritch invocation instead of a feat

So, concerning the eldritch invocation: the main issues when throwing a non-Thrown melee weapon without any invocation or feat are currently that 1) you'll deal significantly reduced damage as opposed to melee, 2) you end your turn without a weapon, preventing you from making attacks of opportunity, and 3) you have to spend an entire Action to get your weapon back.

As a comparison for power, let's take a look at the Improved Pact Weapon invocation (from here on, IPW) from Xanathar's Guide to Everything (p. 57), which gives your Pact of the Blade weapon a +1 to attack and damage rolls, as well as allowing you to summon ranged weapons like bows and crossbows (which is pretty similar to the effect you want to achieve).

Let's deal with the issues listed above:

1. Reduced damage: ranged weapons deal less damage than melee weapons by design, since you get the benefit of being out of range of melee opponents while still damaging them. The IPW invocation allows you to summon a Heavy Crossbow with a +1 to hit and damage rolls, equaling a mean damage of 1d10+1 = 6.5 before modifiers, while the strongest melee weapon (in terms of mean damage) deals 2d6 = 7 (a 1d12 melee weapon also has 6.5 mean damage). Still, the Heavy Crossbow has a greater range, so these are roughly equal in power.
2. We can fix 2) at the same time we fix 3) - by allowing the user to summon their weapon as a bonus action. Now, you can throw your weapon as an Action, then summon it back as a bonus action, meaning that you don't end your turn without a weapon and can therefore make opportunity attacks. Simultaneously, you can attack each turn, without having to waste an Action every second turn. However, an issue is that warlocks will want to use their bonus action frequently already - namely, the Hex spell, unless against stronger high-health targets (where you don't have to shift it as often). Furthermore, you'll run into problems when you also have the Thirsting Blade invocation, since you only have 1 bonus action to summon your weapon, but you have 2 actions to throw it. It also doesn't give you any noticeable benefit as opposed to using an IPW crossbow.
3. As an alternative, we could, therefore, allow you to summon the weapon back as part of your attack. This means you don't end your turn without a weapon, you deal a little more damage than an IPW crossbow, and you can benefit from Thirsting Blade. To compensate for these advantages over IPW, you only have a range of 20/60.

In my opinion, the invocation is now fine as it is, but it would not be overpowered to give the thrown Pact of the Blade weapons a 60/60 range (i.e., 60 feet without disadvantage, but no further at all). Hence, here's the final invocation as I would design it:

Thrown Pact Weapon (feel free to switch this out for something fancier, like Arcing Weapon of Destructive Doom)
When you make an attack as part of the Attack action with a pact melee weapon on your turn, you can throw your weapon at a target, and you can choose to have it return back to the hand you threw it with as part of the same attack. For this attack, your weapon gains the Thrown property with a range of 60/60 if it doesn't already have it (existing ranges are, however, overwritten). If your pact weapon exists within 60 feet but isn't in your hand, you can still make the attack if you choose to have it return to one of your hands afterward.