When you say you replaced the battery, did you use regular alkaline batteries or rechargeables ? Rechargeables are only 1.2V each and alkalines are 1.5V each. Many devices don't care, but I find some remotes can be borderline.
Also, if I recall there was a remote firmware update. If you go to the remote utility, what version of firmware does your remote show ? The latest I think was pushed with the V20250815 firmware last fall to update the BT500 to V7.
Did you try to reset the remote. And test it out in IR mode only for a while to see if it behaves the same. Then go into the bluetooth settings, and "Forget" the device to ensure it is no longer known to the box, turn Bluetooth off for 5 second, and then turn it back on. And then re-pair it to the box ?