Yeah, without knowing the draw on the original strips it's tough to tell what's left in the tank. If you have a killowatt meter or one of those gadgets around we can figure it out close enough.

My intuition is a 350mA boost buck will have plenty of room, but that's me :-)

For the relay driven 24volt BTF I suggest one of these.

This way you turn the strip down if it's too bright. Trust me, those 24volt BTF are stupid bright.