Was looking on the internet for a solution and stumbled upon your post.
Mine has stopped working completely.
Upon initially upgrading to iOS 16, it worked briefly in HomeKit where I can see its status as locked or unlocked, but now only get No Response or This Accessory Is Not Responding.
HomeKey was giving me the same problems too as you described in your post when I first upgraded to 16.
I tried restarting the HomePods and hubs but no success.
I tried removing the lock from HomeKit aka Remove Accessory and now I cannot add it to HomeKit.
I’m just thinking to return to my regular deadbolt I had before.