You can't use more than one Brige with the same account. You would have to re-pair the Brige in the app every time you switch.
But the range is usually sufficient because each lamp acts as a repeater. So maybe you need to put another lamp in the stairwell to control the lamps upstairs.

The Hue lamps and luminaires build up a mash network. The approximate range from one lamp to the next is about 10m.
This means that the system can also be operated on different floors - the important thing is that the last lamp on one floor can establish a connection to the next lamp on the other floor.