I was looking at cap jump packs a few years ago but I like the flashlight, USB and inverter capability of my 770R
My GF has a huge Shumacher jump pack (1200amp I think it is) with a built in AC power inverter, usb ports etc. And she uses it extensively

not only to jump start her dodge minivan and E250 cargo van but when were at the Thompson Motor Speedway swap meet in Conn. back in early November and ran drop lights for her tent sales setup with it I was impressed. Definitely worth the money.
While I was there I scored a jump box myself, an off brand name 1000 Peak amp for only $8.00

"Super Start".
Took a chance it be good was showing a 75% charge when I purchased it, fully charged up fine after I got back home. Plugged in a small 140w power invertor to the 12v port and checked voltage output. Even ran a floor fan

If I do need another battery I figure may be ahead of the game on this one. I also have a small Shumacher 400 Peak I had purchased just for my motorcycles and riding mower.
Some years ago I had a Coleman jump box and it had its own external pin style AC charge power supply which was lost in a move and pretty much rendered that useless. I did take the battery out to charge it, but dead by then.
Really do like the fact all these newer ones just use an extension cord.
To address your earlier question I have no clue who supplies Clore battery somewhere I read they claim its there "own" battery.
As for the ones I linked to AJC seems to be the actual manufacturer and their batteries are sold under numerous "brand" names. As for a giving performance boost I just would think the 22ah would simply provide slightly more reserve cranking power.