Thanks James, I appreciate your frankness and guidance. I put this out there for this very reason. I'm here to learn. Hopefully one day I'll be able to share my experience with others, once I get some..

It finally sunk in that I need to get power from the car battery. I'm going to do that.
The Amp is a Xiegu 125B. Specs:
Frequency Range: 0.5-54 MHz
Maximum Output Power: 1.8-30 MHz >110 Watts / 50 MHz >90W
Maximum Ambient Operation Temp: 131 Fahrenheit(55 Celsius)
Gain: 13 dB (+/-2dB)
Spurious Suppression: >50 dB
Supply Voltage: 12-15 Volts DC
Transmit Current Draw: @ max output 30 Amps
Receive Current Draw: @ max 700 mA
ATU Frequency Range: 1.8-30 MHz & 50-54 MHz
ATU Tuning Range: 14-500 Ohms
So the only question I have remaining is this. Say my car audio buddy gets power from under the hood to under the seat. I'll also make sure he addresses the fusing and grounding needs... So could I put this power into a RigRunner to distribute to both the amp and the radio?
If this would be an issue I could just continue powering the radio the way it is now, off the AC outlet in the console. Running it off the battery would be nice though as this would run around the auto shutoff issue.
Thanks again, I really appreciate your help with this.
Chris
KI6GEA
I just read the manual (gasp!) of my new Tahoe, and learned there is a button right on the dash to disable the auto shutdown. It resets each time the car is shut off, but at least I can disable it for times I'm using the amp. I'm only going to use the amp for the HF antenna I think. I have an antenna duplexer for the vhf/UHF amp...
So I have this little Powerwerx 30amp power supply, I could put that under the seat where I was going to put the battery, and run that off the AC plug in the center console. And run both the amp and the radio off the Powerwerx I think?
I had a loaner vehicle when mine was in the shop that had the shut down option. I was able to go through the dash menu and override that but when I did a manual stop to go into a store and shut the vehicle off I had to reprogram when restarting. Personally I hate the engine shut off feature but that's me.
On most of those systems you can program the vehicle to remain idling when stopped. I'd think that was the easiest route to go, but you may have to reprogram after a complete shutdown.
Your new, so we are going to go easy on you.
BUT you are really really really messing up thinking about putting AC/DC power supplies in your vehicle to run 100 watt amplifiers. And I mean in a big way. Forget ALL of this nonsense, and do the thing you DO NOT want to do. And that is run a power cable under the hood to the battery/ground as Glen outlined and start hooking up these DC appliances (rig and amp) up to DC. You re not going to be able to pull the amount of current you need to run a 100 watt amp with AC to DC conversion from some wimpy little AC inverter in your vehicle. You can get away with it with your 10 watt transceiver, although on general principals even that is a power wasting double conversion solution, not even to mention the noise spikes inverters might put on your power line. In other words, this just isn't done. Not a good idea. The proper solution to almost ANY problem is to always bite the bullet and do that thing you do not want to do.
Now lets hear more about this QRP to 100 watt amplifier you intend to run with your 705. Make, model, and specifications. Please don't be turned off by my "frank" analysis, I am trying to set you on the proper path. You are now in the ditch, but we all are here to help pull you out. 