An external battery pack for the //c is up on fleabay
I don't see a way to recharge the battery.
Apple //c PSU -> Power Mat -> Apple //c
Had one... it was a POS.
When I saw that, I figured the battery inside is probably dead dead dead NiCad job, and had no urge to create a luggable IIc.
That's what the Prairie Power Pack was for.
Baaaack when I had a Power Mat, I used it as a UPS for my Apple IIc. It wasn't cost effective for long - cheaper, more capable UPS units were coming out.