I use an inverter tied to my RV house batteries. It passes through the 110V from the contest, but switches to the RV if the external power fails. It got a "field test" recently when the power went out 4 times at a contest. It worked flawlessly. This kind of setup would also work with just a regular car battery, though I would recommend getting a seperate deep cycle battery for this purpose to you can still start your car if there is a long outage.
I have also used a Duracell power pack succesfully. Just be sure it's fully charged first. It takes a long time to charge, and the charger that comes with it doesn't put out much power, so it won't recharge the battery fast enough keep up with your load if you start out with a power pack that isn't fully charged.
I have tried computer UPS systems in the past, and I don't recommend them. The reason is that a lot of contests use generators, and some generators put out a square sine wave for power that your UPS may not recognize. Your UPS is looking for perfect power from the power company, and it will treat some generator power the same as it would a power failure and just run off the battery. My APC UPS has a sensitivity adjustment for generators, but even so I have found some contests where it just doesn't work. Atfer having this happen a few times, I retired it and got the inverter.
In regard to the sleep question: Get a Maverick ET-372 thermometer and set the low temp alarm. If your smoker temp gets too low, it will wake you up.