Unfortunately there aren't any good cordless radar detectors. All of the options available feature very outdated technology. Additionally, in order to conserve battery power, the detectors are constantly going to sleep briefly while you drive which hurts detection.
The best one used to be the Escort Solo S4, but its underlying platform hasn't been updated since the 90's and it's since been discontinued. Performance was really bad, there was no GPS, it would misreport K band as Ka, it lacked modern BSM filtering so it falsed constantly, etc. So even the best option was really bad...
Sticking to a cig. lighter for power is the preferred option for rental cars, though I understand that's not always ideal. If you don't want a cig. lighter cable present in a permanent install, hardwiring is the way to go.