I am sorry but there is no such thing as a 2A or any amp battery. There is
amp-hour rating symbolised as
A·h. When you say 2 I assume you mean 2Ah with the 12volt battery. It does not mean you can use 2A maximum. You can draw 5Amps from the battery if you want.
Some more information can help
- What revision board is it?
- Do you plug the stick in when the Pi is on? Does it reboot?
- Do you have it plugged in and does the Pi boot normally- not safe mode?
- What is the 1117 amp rating?
Like you said the Pi uses maximum 1A which which should at the worst case give you 2 hours battery time. Now why the peripherals do not turn on might be caused by the line converter you are using. Looking at a generic spec sheet of a LM1117 it is rated for 800ma - These things are smart now a days with current limiting, short circuit protection, etc.
What you can be experiencing is the current limiting feature on the regulator? Maybe the peripherals does try and turn on but because it does not have enough current the internal circuit of the peripheral gives up and does not turn on again (like a polyfuse) so you have to take it out and try again. It happens so quick that the Pi does not experience a dip and does not restart but too quick for you to notice anything happen.