The rebreather will last several hours on a single fill depending on the model, cylinder size, scrubber size, etc. Unfortunately it may not solve your problem. You will need to fill the oxygen cylinder from a larger oxygen cylinder either via transfilling or with an oxygen booster(which requires around 150psi of compressed air to drive it). Obviously with transfilling your fills are limited to the pressure in the larger tank. You will also need compressed air(or whatever mix you choose) in your diluent cylinder. These rebreather cylinders are around 19cuft typically so it might be practical to bring several with you.
It's possible there might be some very simple and inexpensive solutions to mitigate the rebreather shortcomings considered in your response. Not all rebreathers rely on the use of AL19s, you could very easily choose onboard tanks up to LP27s which add about 45% more available on-board gas assuming you don't live in cave country. While 27cu/ft might not seem like a lot of gas, I would be willing to bet that with regulation fills (i.e. 2640) you could realistically get many hours with that much gas as long as you're not going up and down (rinse, later, repeat).
The use of a transfill whip is actually quite a bit more practical than it might initially seem. Often when I dive my CCR rig in barely-CCR-friendly locations I actually make arrangements for an Aluminum 80 of Diluient (say Tx21/35) and an AL80 of Oxygen to be present. Granted, I can't defy physics and totally refill my AL13s to 3,000 by transfilling them from the AL80 source tanks, but when you only need a couple cubic feet per hour of diving you reach a point where you can do one hell of a lot of diving in this arrangement. Who cares if you get a 150bar fill on the O2 side when that short fill still provides as much as 3 or 4 hours of diving. In fact, more than one dive shop around won't boost O2 to 3,000+ psi, so you'll be getting a regulation fill in some neck of the woods. Even a 100bar fill on a AL13 is well, enough to go see some stuff. There are options available with the diluient as well, in that you could use best mix of Enriched Air (low enough that you can still blow down the PO2 when necessary) which will allow you to rely less on the O2 cylinder for breathable molecules (popular approach with a few cave country instructors).
The other consideration worth mentioning is the use of off-board gas. While it's true that popular CCR tanks like AL13s, AL19s, and LP27s only hold a small amount of gas, the use of off-board gas turns a set of AL40s or AL80s into an awesome setup for some stay away from the compressor diving.
One possible configuration is mount your standard onboard tanks, and then connect off-board AL40 of breathable diluient
, and connect off-board 100% O2
and you'd have to introduce yourself to the LDS owner upon your next return. Some may caution that you'd need to carefully consider your bailout and decompression requirement such that you don't use so much diluient you can't make it back to the surface - and they'd be right.
But, with a little planning and some idea of what you're trying to do (a simple academic exercise to most CCR divers) you could configure for days of diving and only change scrubbers and perhaps move a little bit of gas around with a a whip.
In any event, the machines aren't so rigid that you have 13 or 19 cubic feet of gas and then transform into a pumpkin - there are a lot of options available to keep you away from the LDS for extended periods without hauling expedition levels of equipment.
I'm just throwing out some things to consider, especially since the rebreather can go as carry-on luggage to fascinating diving destinations far and wide.
