I have a cigarette lighter plug attached to two of the radios, the ones that don't transmit. The one that does transmit I use as a base and it is powered by an ICT Comm Series power supply 13.8 VDC. The two that won't transmit are powered by a different power supply that measures at 13.8 as well. Tested all three with the same antenna. Guess the next thing to do is test the two on the same exact power source as the working radio. I appreciate all the feedback and will let you guys know what happens.
Cigarette lighter sockets/plugs can be problematic at best.
They will supply 12 volts to the radio, but often they are fused at 10 amps, which might be below what the radio will pull at full power. That usually blows the fuse, though.
The spring contacts don't always work well, either. It might make contact enough to power the radio, but when transmitting, it'll pull higher current and result in a voltage sag. That voltage sag/drop will result in the radio restarting.
The wiring for cigarette lighter plugs is usually bundled in with all the other circuits in the vehicle, that can include CAN Bus stuff, or other noisy circuits.
You'll be doing yourself a favor by ditching those cigarette lighter plugs on the radio and wiring directly to the battery as Kenwood recommends. There's a good reason that's what they tell you to do.
The antenna can also be an issue. Make sure the antenna is outside the vehicle and mounted properly. Make sure the antenna is properly tuned for the frequencies you are using. High SWR or a high RF field near the radio can result in issues when transmitting.
Trying out those two radios on your base power supply would be a good troubleshooting step. It'll tell you if it's a fault with the radio or the power/RF.