Quick update - I added a 1A fuse between the battery and the USB charging and protection board, and that is how I plan to leave it. I took these measurements after installing the fuse to see what voltages the charging protection board would cut off the battery:
Battery Low Cutoff 3.04v (with 1A fuse in series with battery)
Battery After Charge 4.22v (with 1A fuse in series with battery)
These work for me, but I am still not really happy with the performance. I tried a 5V regulator board and a 5V fan but the performance was only modestly improved, if that. I think I need to go bigger on the fan to have adequate solder fume extraction, but this will work in a pinch.
Lithium Battery TP4056 DW01A Charger Board Module
3.5A DC-DC Step-up Boost Voltage Converter Module 3V 3.3V 3.7V to 5V 6V 9V 12V
(could not find the exact link but there are tons of these tiny boost controllers on Ebay)
I used the 12V version, the voltage should match the fan being used.
FBApluto 15pcs Female Micro USB to DIP 5-Pin Pinboard 2.54mm Micro USB Type Interface Power Adapter Board 5V Breakout Module (I used these for the fan and battery connections)
https://www.amazon.com/gp/product/B07KS1RPMP
3A/250V 6A/250V switch:
(Pack of 40 Pieces) MCIGICM AC 3A/250V 6A/250V 10A/125V 2 Solder Lug SPST On/Off Mini Boat Rocker 2pin Toggle Switch Car Auto Boat Round spst Rocker Switch
https://www.amazon.com/gp/product/B07D7NCLKF
Gikfun 12x12x7.3 mm Tact Tactile Push Button Momentary SMD PCB Switch with Cap for Arduino (Pack of 25pcs) AE1027
https://www.amazon.com/gp/product/B01E38OS7K
nidici 20pcs(10 Pairs) 1S Lipo Charger 3.7v Battery Charging Cable Male & Female for RC Quadcopter Drone Multirotors
https://www.amazon.com/gp/product/B01MXE9IE8
Carbon filters (used these), the filters linked below are for garbage can deodorizers.
https://www.amazon.com/gp/product/B00XNI5SY4