Bluetooth disconnecting when iPad sleeps: apple pencil not reconnecting

Hi. Even since the most recent update to IOS, my apple pencil disconnects from my iPad when the iPad sleeps. I've noticed the following patterns:

Ipad sleeps for less than a few hours (one minute of sleep to a few hours of sleep): Bluetooth won't recognize the pencil. I have to go to settings/Bluetooth. Flip Bluetooth off, then back on, and then the apple pencil is recognized. Through this process, when I go to settings/Bluetooth, the apple pencil doesn't show for a few seconds, then pops up as disconnected. I again toggle the Bluetooth off, then back on, and it shows and connects.

Ipad sleeps for a long period of time (most likely overnight or several hours): When I turn on the iPad after a long sleep, the iPad recognizes the pencil and I don't have to toggle the Bluetooth on and off.

I've also noticed that if I'm quick after waking up the iPad to get to settings, Bluetooth shows unavailable /grayed out for a few seconds, then it comes back to life.

I have left the pencil attached to the iPad to update the pencil software.

I have disconnected the pencil completely, shut down the iPad, started again, and repaired it (I tried this a few times).

I have a 1st generation pencil Firmware 02.61.00 Hardware 01.00.00

Ipad Pro 12.9 inch, software 15.3.1.

I've noticed the same thing happening on my phone. I have a fossil hybrid watch, and it has the same issues staying connected to the phone, so I don't think the issue is isolated to the iPad/pencil. It feels like a Bluetooth/sleep/wake issue.