Allow a hefty normal capacity (for various reason often to do with gun/tactic combinations to experiment with)

But maybe give some well define bonus's for keeping a light load :- to make it a goal worth pursuing

Faster movement/maneuverbility, quicker reactions, lower exhaustion rate, fitting thu smaller holes, being able to go prone, etc...

Sufficient availability of required tools/resources to 'grab' for specialized situation would be required (and sometimes is a 'dead give away' that such situations are approaching)

Player would have option of when to discard for advantage or to hoard inventory/tools

Give access through a quest or whatnot, that rewards the player with some sort of "mule". This "mule" will come to the player when called/summoned/is always following the player or any other reason you could get it to you.

The point to the "mule" is for it to carry anything you give it, to your house/HQ/base/robbed bed and box. This mule could be a summoned creature or demonic shopkeep (poke at skyrim), a bird or pack dog to be called, or an actual mule that follows you around and waits outside dungeons. The downside to followers that every player knows is that they have a penchant for getting themselves killed (by enemies or friendly fire), not to mention bad pathfinding that runs them off cliffs or "the long way around".

There was a diablo clone that you had a pet that would do something similar and when you sent it to town you would have to wait till your pet got back.

A summon would have the advantage of a one way trip but with lower capacity.

This solves the inventory game while your adventuring and postpones it until you get home.