Actually I missed a very important part of that out - I need a stable 5v to run the Nano - if the battery didn't go all the way up to 4v one could run the nano straight off it... so right now I have a power supply that puts out under 2 amps, shuts itself off at 2.9v battery but can't seem to come up under full load when the power comes back on - so I'm using the NANO to shut off the Pi before the battery gets that low - and warning the Pi in the process with an output so it can shut down gracefully - then - when the battery gets all the way back to around 3.7v I turn the main output on and the Pi starts up - this way it is impossible for the Pi to get caught out or end up stuck in the off state.... but I'm using the 5v to power the Nano and the 5v relay (as they use MUCH less power, the supply will start from a flat battery and provide 5v when it is able.
In a wonderful world I'd have 2 5v outputs - one very low power able to run as soon as the battery can handle it - the other under logic control that the NANO could bring in when it decided the battery had enough power.
So the things that seem to be wrong with many of the supplies out there- include one or more of the following:
(a) not enough output - i.e. 1amp
(b) inability to provide both output and simultaneous charge
(c) inability to go from flat battery to turning on full output without the output being disconnected at first.
(d) cost prohibitive
The control side with or without display - easy - the bits above - not so easy for me.
Not quite the simple answer I guess you asked for