A 7 to 10 A-H sized battery should suffice. Or just match the physical size. The starter draws about 8-10 amps.
You can check if battery is being charged with a voltmeter on battery. The battery should rise above 13.5 vdc with engine running.
A fully charged battery with no load, rested for a couple hours, should be at 12.7 vdc.
Problem with electric start generators is they sit for months at a time without charging the battery. The battery self discharges and become sulfated so it can not be recharged.
I use a 'Battery Tender' on my Yamaha EF3000i to keep the battery at 13.5 vdc float voltage.