Consider making the tile manipulation a Lair Action.
Lair actions are a method other than Legendary Actions to give a monster ways to balance the action economy between themselves and the party of Player Characters. They usually pertain to altering or using the terrain of their lair, and can only be performed by the monster (or NPC) when they are in their lair. You've mentioned that the party will be facing this BBEG in their lair, so it would make sense here. The benefit of making the manipulation of the anti-magic tiles a lair action would be that the BBEG will still have full use of their action, bonus action, and object interaction on their turn, not needing to waste any of them.
As an example, an Ancient Red Dragon has the following Lair Action options, and chooses one to perform on initiative count 20 (losing initiative ties) each round.
- Magma erupts from a point on the ground the dragon can see within 120
feet of it, creating a 20-foot-high, 5-foot-radius geyser. Each
creature in the geyser’s area must make a DC 15 Dexterity saving
throw, taking 21 (6d6) fire damage on a failed save, or half as much
damage on a successful one.
- A tremor shakes the lair in a 60-foot
radius around the dragon. Each creature other than the dragon on the
ground in that area must succeed on a DC 15 Dexterity saving throw or
be knocked prone.
- Volcanic gases form a cloud in a 20-foot-radius
sphere centered on a point the dragon can see within 120 feet of it.
The sphere spreads around corners, and its area is lightly obscured.
It lasts until initiative count 20 on the next round. Each creature
that starts its turn in the cloud must succeed on a DC 13 Constitution
saving throw or be poisoned until the end of its turn. While poisoned
in this way, a creature is incapacitated.
Because you know you want your BBEG to manipulate the antimagic tiles every turn, you could simply limit them to one option for their lair action: to manipulate the tiles as they see fit.