Can I drop 4.5 V to 3.7 V with a diode?
Let me answer some basic misconceptions that you may have before going too far:

The difference between button cells and LiPo's is extreme; the current they can deliver is just about as far apart as you can get (few milliamps for a button cell, dozens of amps for some LiPo) Substituting one for another sounds like a bad idea. 

Additionally, a "3.7 V" lithium-polymer battery has an enormous voltage change as it discharges, how fast, and at what temperature:




Source: http://www.rfcafe.com/references/electrical/battery-li-poly.htm