Simple check if your not up for putting a meter in series is to check for anything running hot. My math says 1.5 Amp hours in 45 minutes is 2 amps draw, at 3.7v is 7.4 watts. Given likely consumption for the Teensy+ screen is 0.1amps finding the part that gets the hotest may be a low tech way to do this.
Also potentially useful is a lipo fuel gauge that tracks actual current vs current out, at least for your prototype
https://www.sparkfun.com/products/10617
Though I think I've seen better ones that actually track current. Useful since if you start doing sleep and similar things current draw will pulse and be hard to track on a meter.
Given the limited information you've given (no actual measurments on your system) my guess is that your boost converter isn't doing a very good job, and most likely neither is the LED power circuit in the display. The LED test could be as simple as loading code that writes to EEprom once a minute to increment a counter and leaving it to run with and without the LCD backlight connected.
With the power supply that last paragraph needs some answers to get your mental model in a better position.
If you can't supply enough current to the boost converter it can't make more current, the output voltage will droop and/or the PSU shuts down.
If your battery can't provide the current that the boost converter needs, it's voltage will droop and may hit the point where it's protection circuit trips and the whole thing dies (so possible you have a surge load that's dropping out a battery still at half charge, though your design should have no surge loads at the moment).
Boost converters can vary widely in efficiency, and that efficiency only applies at one current, and that current may not be Imax. If you can't get a I vs effeciency graph then that's a warning sign.
You don't care what the total capacity of the boost converter is, you want the one that is operating at 70-80% while there, hopefully with some head room. So dropping a 3A unit in there may not help, unless it's good at feeding <300mA loads.
And if you boost to 5V you will still be wasteing 1/3 of the battery capacity in the step down regulators on the Teensy and LCD, though at the moment that 60 or so mA is not a big part of the loses.