High Current Battery Power
Since you said cheap........
Any car repair place can get you a used car battery that has one weak cell. Sometimes just for the asking. Car batteries get 12V from 6 2V cells in series. So charging it at 13.8V yields a 10V output. A simple buck converter fed from this setup will get far beyond 20 minutes, even with full load demand. The buck converter will tolerate batteries that get worse for awhile as well.
If that requires too much weight or size for your application then consider feeding the buck converter from a 12V 12AH lead acid cell used to power UPS. Be careful to use an off the shelf charger, or provide some temperature sensing to limit charging current if you build your own charger. 
Your full load demand could make the battery's internal temperature rise so it will not tolerate as much charging current.