When I’m mounted, can I use the Bait and Switch maneuver exchanging the mount and my ally’s position?

Tasha’s Cauldron of Everything added one of my favourite maneuvers (Bait and Switch) so in my game as a player I used it while mounted and the DM was like "Nice, like in chess (The king and rook maneuver)". But I was thinking, is possible, rules as written, that me and my mount both change places with my ally?

Bait and Switch: When you’re with in 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 5ft of movement and the creature is willing and isn’t incapacitated. This movement doesn’t provoke opportunity attacks. (…)

Is it up to the DM? Because it doesn’t specify.