I sympathise with the OP, and also agree with many of the comments already made about the proposal. Largely, the merits of adding a 3rd spell slot for a warlock at 5th level will be table-dependent. I can see where it may make a substantial difference to the power of the class, and equally where it will make little difference whatsoever.
If you were going to do this, however, I'd be inclined to shift the additional slot to 6th level, rather than 5th, as the warlock gets 3rd level slots at 5th, a proficiency bonus boost at 5th, and a eldritch invocation at 5th. At 6th, it would add to the warlock patron feature, but at least it spreads out the bennies a bit more (and see my suggestion below about making it an alternative feature to the 6th level warlock patron feature).
Nevertheless, the warlock class is not really one of those classes I'd put in the top tier of powergaming. Its principal powergaming abuse is as a 1-to-3-level multiclass dip for another Cha-based class. A single-classed warlock is generally not as flexible as a single-class caster of another spellcasting class (Tomelock, maybe?), nor as lethal in combat as a single-class nonspellcaster. In that light, I suspect adding a 3rd spell slot at 5th level (and presumably a 4th a 11th level?) won't make it that much more powerful, and might temper the characterisation of the class as an eldritch blast-cannon one-trick pony.
I wouldn't be a big fan of making additional spellcasting in the way you've described as just another invocation, as warlocks don't really have enough invocations anyway. In most campaigns, a single-class warlock is probably going to top out at 5 or 6 invocations, and of course will be playing for much of the campaign with less (i.e. most campaigns tend to end before the characters achieve 12th level). However, it would present a bit of an interesting character design choice for the player, so worth thinking about.
Another option along a similar vein would be to have an alternative warlock class ability as per Tasha's Cauldron of Everything, which substitutes for the warlock's 6th level warlock patron feature. At 6th level, the warlock could substitute their warlock patron feature of that level with a "patron spell slot" that can only be used to cast a spell from the patron's spell list (but otherwise is treated as any other warlock spell slot).
Cheers, Al'kelhar