A method and system for controlling connectivity of a user equipment device (UE). When the UE is connected with an access node in a cell provided by the access node, the access node determines that uplink noise in the cell is threshold high. And responsive to at least determining that uplink noise in the cell is at least threshold high, the access node applies a battery-level-based UE-offloading process to offload the UE from the cell. For instance, the access node could cause the UE to hand over or otherwise transition from being connected in the cell to instead being connected in another cell.