Same issue here, and my HVAC guy told me an easy fix. The G wire on the Nest is only for manual fan control. If you don’t need that, you can repurpose it as a C wire.
Turn off the furnace, move the G wire to common at the furnace, then move it to C on the thermostat. Turn the furnace back on. Now, your Nest will get constant power, and you won’t have to deal with battery issues anymore.
If you don’t feel comfortable doing this, a tech can do it for under $100, which is cheaper than replacing the thermostat.