Okay, maybe I had an incomplete picture.
I looked at benchmarks for home-built proton packs, which almost certainly consume more power.
Based on how long people say theirs last on different batteries, I came up with an estimate for a Proton Pack of 1.7 amps of current.
I plugged that into the following calculator with voltage of 4.3 volts...
https://www.rapidtables.com/calc/electr ... lator.html That yielded about 7 watts.
A 3.7 volt 20,000 mAh battery would have a capacity of 74 watt hours.
That results in 10.6 hours roughly but you want to reduce that down to about 85% due to inefficiency.
So a 20,000 mAh battery would power a proton pack for around 9 hours.
I got looking and my concern with most battery packs I own that could fit in the side compartment is that they tend to shut off when there's no draw. That could be a hassle.
So I looked for "always on" power banks and I found that the company Voltaic makes some.
They have a V25 rated at 6,400 mAh for around $40 and a V50 rated at 12,800 mAh for around $65. Their V75 is rated at 19,200 for $79 but is too big, at 6 inches. I'm not sure if the V50 will fit at 4.5 inches.
The V25 is 3 inches by 3 inches, which should fit. It's a bit pricey at $40 for 6,400 mAh but I like that it's always on when plugged in (no off switch or timeout) and passthrough, so it could work with a wall charger as a USB power input.
I suppose you could keep a second, cheaper USB battery in there too, attach it to the V25, and essentially have 12 hours of continuous power (9 pass through running from the other battery through the V25 and 3 hours direct from the V25).