SensorPush BLE sensor battery reading

I’m pretty happy with my SensorPush outdoor temperature/humidity/pressure sensor. It’s not cheap but it is calibrated to work in a Canadian winter (most of it, at least). Their BLE adverts are automatically detected by HomeAssistant and decoded. Sometimes it appears to have a bit of a nap, adverts stop arriving (and I’ve checked with other BLE receivers) until the Android app wakes it up, probably the direct Bluetooth connection is all it wants. Also, the battery voltage doesn’t appear to be broadcast. SensorPush do actually publish an API but they only show the GATT characteristics that require active scanning. Also my readings don’t agree with their statement that I can read battery temperature. The Android app happily returns 20.9C (which seems high!), raw values seem to float around 0xffff. So, rather than try to get an ESP32 to lock up its active connection, draining the battery to see how quickly the battery is draining, I’m going for something more intermittent. ...