I'm having problems iwth iOS 10.3 beta 2-4 with sluggish and laggy phone response esp when battery is low. It seems to go away once the phone is charged. Also on iOS 10.2.1 I had problems with the battery gauge being off, those seem better in iOS 10.3 (the phone will correctly shut down at 1-3%, but still when restarted battery is 30% and quickly jumps to 60-70% with charge).
ios 10.3 beta sluggish lag and battery life
I am also experiencing these things! It's making me want to restore back to the previous beta. I don't know how to reduce the lag and it is quite annoying, although like you said, plugging it in seems to improve it a little bit.
Very routine for battery management protocols to fluctuate during betas.
Very bad bettery life on iPhone 6s, iOS 10.3 beta 5