I've calculated my Arduino Uno setup to be ~2200mAh (this includes LED and some sensors).
What would be the best rechargeable battery setup for a 4-6 hour run time?
LiPo seem to be the way to go, however the larger LiPo batteries are too bulk for this project. Also not familiar on how to run them with a switch, as I've been told you can only directly disconnect and reconnect them. Switch still drain the battery.
9V have too low a mAh, but conventional batteries like AA have too low a voltage (cumbersome to charge numerous AA's).
mAh is CAPACITY not current. Perhaps you mean it uses 2200mA or 2.2A? Also you haven't said what voltage is needed.
But anyway you have your answer. If it uses 2.2A and you want it to run for 4 hours it needs 2.2 x 4 = 8.8Ah. Six hours would be 13.2Ah. Both at whatever your voltage is.
BTW a Lipo, like any other battery, can be connected through a switch. Whoever told you it couldn't didn't know what they were talking about (or were asked the wrong question).
If a LiPo is too bulky, there's no better alternative for energy density...
And if your circuit uses 8.8Ah then the battery will need to be speced more like 15Ah because
nominal battery capacity is only valid for a brand new battery, over-charged at the right temperature,
in laboratory conditions - real batteries always lose capacity with age, have lower capacity at different
temperatures, and less capacity at higher current drains.
Batteries that are discharged 100% have much much shorter lives too - hence the factor of two
in capacity headroom I suggest.
Any battery that gets over-discharged even once, briefly, will also lose a lot of capacity
too, so you may need to consider protecting against accidental over-discharge.
Read and read again everything in reply #4 above. It is all very accurate.
@otter, what is it that you are doing?
We (kind of) boiled down your information to 2.2A and a desired duration of 4-6 hours.
What voltage? What application?