Are both hives environments the same? Same upper closure? Insulated? The pics look like different tops? (one is taller than the other?) Overwintering with different lids could account for the differences between the two? And you're keeping the double deep brood setup?

You can force her down by reversing the brood boxes.

Some say the bees will figure it out and the brood area will extend downwards naturally. You will likely want to force her down so you get your nice honey umbrella in the upper box for them to move into next winter. Left to their own design I imagine the bees will want some sealed honey/comb below for separation from the entrance. I think this is where the advantage of a slatted rack type bottom board becomes evident. With that type of bottom the queen will often lay solidly right down to the bottom of the frames in the bottom box. Especially with a screened bottom, I expect the bees will be less likely to move down on their own.

Some checkerboard to encourage the cluster to move.

Some help huh? So you feel compelled to intrude. OK I look at the bees then and checkerboarding will be more disruptive to the daily hive activity because you are moving stuff around in the brood area. If you just swap boxes (reverse) that preserves the integrity of the brood area (order of the frames) is much less disruptive to daily bee life and you will be happy that the queen is where you think she should be.  :^)

I look at it this way I try to intervene least as possible in the bees business, because bees will be bees and do what they dang please! You can try to steer them some with attractive opportunities, but sometimes they give you the finger. Especially if you are mucking about in their life.

Solid bottom or screened? I'd say a screened bottom should not be open or un-stuffed yet either. I'd probably unstack the boxes check my hive stand and clean the bottom board (inspect it closely it may have to support a couple hundred pounds of bee hive) and then reverse the boxes. Leave the sugar on above the cluster til later in case of a cold snap and put the lid back on it. Ready for the new season. I have trees and bushes budding out and even some blooming. First dandelion bloom yesterday.

When they get that top box filled then add your excluder and a super. But not yet, I have had trouble giving them too much space. I think we still have some weather coming, I give it 3 weeks to 30 days more of wet. If it's good they might fill that box quicker than 3 weeks but not likely. You will want to add the supers one at a time as they fill the box below. Problems can arise if you give them too much space. The excluder stays on top of the brood boxes below your first super, it keeps the baby bees out of your honey.

For a population booster for the small one you could give them a pollen patty. Bee judicious in monitoring population when applying patties early, they can bust out. In a double deep brood setup less likely to bust out.

If the weak colony is too small, use a double screen board to separate the colonies and put the weak box on top of the stronger colony so it will be heated. However don't share the mites if its possible either is a mite bomb.

Re-reading the "deep end" its not clear, sounds like you are wanting to switch to medium brood boxes? In that case, removing the lower box is prudent as the bees have likely moved up out of the lower box. Then place your new medium brood box above for them to build into. Then this time next year do the same and you're out of the deeps. If you add medium frames to a deep box they will build comb (usually drone cells) off the bottom of the medium frame.