You can choose to bring the grappled creature with you, but it takes 10 feet of movement.
Bait and Switch says:
When you’re within 5 feet of a creature on your turn, you can expend one superiority die and switch places with that creature, provided you spend at least 5 feet of movement
When you use Bait and Switch, you are using your available movement, so ...
The squeezing rules are not about passing through narrow openings
The squeezing rules are about movement in combat, nothing more. They are specifically in the Combat section of the rules.
To adjudicate a creature passing through a small opening or tunnel, the DM has to make a ruling: doable, impossible or doable with an ability check.
Ask the GM, the rules are not precise and exacting
There are no exacting dimensions on the biggest medium creature or the smallest small creature and whether the first could squeeze into a space the size of the latter. You'll have to ask the GM. That said, there are rules:
If you are not using a grid
Then a creature can squeeze into a space that it ...
The illusion must be no larger than 10 feet a side - but can move with the target
The spell description (emphasis mine) requires that the created phantasm must be of a size no larger than 10 feet a side, thus, a corridor cannot look "infinite":
You craft an illusion that takes root in the mind of a creature that you can see within range. The ...