Ken,
I don't have the link right handy but for whatever generator you end up with, you can get a converter for it that allows the generator to run on gasoline, propane or natural gas. I have the converter on my 8550W unit and it works but it is a bit hard to start on propane when it's cold. So I start it on gas, then switch to propane. Reason I bring that up is you'll get a longer run time than what you get on 5 gal of gas on the typical home generator. That gets you about 12hrs of run time with a 50% load. a 20lb bottle of propane gives you a little less run time but.... You can get larger propane tanks that would give you plenty of run time.
You could get one like this that has an LP option - with electric start capability
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all you need is a trunk monkey that jumps out to hit the start button when the electricty goes out..... You can find SCADA type of equipment that will do that by sensing when the power goes out and trips a relay to close a contact that starts the electric starter on the gen. But it needs to be smart. Needs to stop "pushing the starter button" so to speak, once it senses the engine starts. Plus, what will it do if it doesn't sense the engine started plus, how long does it wait when the house power is lost before starting the generator and what does it do when it senses the power is restored. So you probably could jerry-rig several items together but may be best just to bite the bullet and get a small whole house unit with the all the bells and whistles (LP, gas, natural gas capable), auto-start, auto-sensing / switchover. Plus, even when you are home and the power goes out - you will have power instead of getting your flashlight out.
A battery backup sump pump with a battery to give you days of run time (on / off cycling) will probably go for about $3k. One house in our neighborhood has one - and it did work while she was out of town last year for a week, so it was money well spent according to her. Only to have her basement flooded (us too...) a month later.
Justify the cost by esitimating the damage caused if the sump pump doesn't work while you're gone. Add in freezing pipes, lost food in the fridge and freezer and whatever else your losses could be if you're not around.
Bob S.