Easiest (or cheapest) way to get both 5V and 3.3V on one circuit
One approach (not necessarily the best) would be to use a boost converter module with some cell such as a (protected) 18650 which is appropriate to your LED loads. One common chip that the modules are based upon is the MT3608. A synchronous chip might be be a bit more effecient, especially near the end of discharge. 

The rest of the circuit is relatively low current, as I understand it, so you could use a linear LDO from the +5V rail for the 3.3V. 

One approach could be to re-purpose an inexpensive USB power bank that uses a single cell. That gets you the cell, protection, charging and 5V boost. Then you just add a little LDO regulator to get 3.3V. I have a couple that look like lipstick cases, hardly much bigger than the cell itself.