Current source with no voltage drop
You could probably make a constant current source with a fet and cheap opamp, but there are cheap ICs (note the example is not, just one I picked out from here) that will do what you need, like the LT1932 (datasheet) which will work from 1V to 10V input (I selected a low min voltage version due to your battery - but of course you can run from the 3.6V stepped up rail), and supply up to 40mA constant current.