Unless I'm very tired this morning, you could install a bazillion kVA transformer and you're not going to create a 240V 3Φ 400A service on the secondary if you're truly limited to 208V 400A on the primary side.
Bob accurately sized a full winding isolating transformer size that could be used but you should note that the maximum secondary current would be 347A on the secondary with 400A on the 208V primary side.
I thought you were asking for a boost configuration/size that could be used. I calculated ~ 11kVA per Φ or 34kVA 3Φ but you should check my numbers and you would want some safety factor so maybe use something in the 45kVA range. Keep in mind you still only have 347A capability on the secondary side. You also now have the problem where do you wire your 120 receptacles etc. to?
I don't know how much load you're talking about for the beds, but maybe a potential solution is one boost autotransformer supplying a smaller "bed panel" with 240V and the rest of the facility supplied from your given 208Y/120 supply? This would eliminate the boost at every bed which the owner seems to desire?
Oops, looks like Augie and I were thinking similar and he typed faster....