You want a back up for your backup generator?
The main reason your lights would go out is the failure of your backup generator to start or it breaking down......not a sudden shortage of natural gas to fuel it.
To cover yourself if the generator fails to start or breaks down, I would propose a battery backup, such as a Tesla Powerwall. One stores about 20 KWh and would instantaneously deliver electricity while you try to get your generator functioning again. I have a bank of 2 Powerwalls, and when utility power goes out, the load transfers to the batteries almost instantaneously. Lights don't even flicker.
Your concern reminds me of my mother, who was fearful of open water, and therefore, fearful of travel on boats and ships. I think she had seen too many Titanic movies. She used to say that she wouldn't go on any form of ocean travel until the lifeboats had their own lifeboats.