I'm just not sure AAA pitching can challenge Schwarber enough. Just as an example here, you're talking about guys like Aaron Brooks, Seth Frankoff, Williams Perez, Jake Buchanan and Casey Kelly being the cubs top 5 in terms of games started in AAA. Of those 5 only Frankoff has an ERA under 4.50. Now that's not the most robust sample or anything but it goes to the point that AAA pitching is no where near the same level as MLB pitching. With a few exceptions, take whatever your teams #5 starter is and add a run or so and that's generally the ERA you're talking about on AAA pitching because they are your 7th-8th type options if a starter goes down.
This is legitimately why the whole "send <x> to AAA" thing really doesn't work as you're describing it. The gap between AAA to MLB is probably as big as the gap between A and AA if not bigger. I mentioned Candelario but another example is 2015 Baez. He hit .324/.385/.527 with a 144 wRC+ in AAA in 2015 and .289/.325/.408 with a 98 wRC+. In other words, he was a 46% better hitter vs AAA pitching. Often times guys just gotta take their lumps vs MLB pitching until they get it or they flame out.