Obviously the cam will have to be battery-powered so consider a dashcam for a car.
Most are powered by a plug-in cigar lighter adapter ("CLA") that converts the car's 12VDC to 5VDC for the dashcam. Because the cord is USB, you could plug the cam instead into into a 5VDC rechargeable power bank.
The dashcam records to a micro SD card. Many newer dashcams use Bluetooth to play recorded video to a smartphone.
The only hitch I can think of right now in this suggestion is regarding live viewing from the cam to your phone...not so sure about live, real-time viewing from the cam to the phone.
=OR=
There are smartphone apps to convert an unused smartphone into a webcam that can stream live video to another smartphone, I saw one on IPCT not long ago. I'll see if I caa find it and come back here to post a link. Or maybe someone else will read this and post it. Then we can both find out if the phone also records an/or if it uses Bluetooth or Wi-Fi to connect to the receiving phone (which could throw a wrench in the concept).