I couldn't say that DHCP is the only thing that takes tiime on the router. It has it's own boot process and that takes time. Is it also set to look for a WAN?
My hotspot is pretty much on 24/7. It has an internal battery and a sleep mode when nobody is actively using network. It charges when car comes on. If for some reason its battery runs low, I can just bring it in the house and charge it. But this is rare.
Idea: get a small gel cell battery, connect it to supply your router and video server when car is not running and charge when car is running (IE via ignition wire and a relay). If your router takes 12V then no inverter is necessary. I forget what voltage the camera server requires but I seem to recall it was 9V so the adapter may be necessary. As mentioned before, a small inverter would be fine if you have to use the adapters. So now, if the gel battery runs down, no big worries, and the relay will ensure it doesn't back-feed the other items on the ignition circuit. And obviously the bigger battery you can get means the longer it will last. If you get a 14ah battery, and the devices are drawing about 1amp it will last "approximately" 14 hours in a perfect world. In reality it will last less.
It's just an idea, not sure how you might want to handle it.