About replacement of 3x AAA by 1x Lithium:
AAA batteries can be either Zinc-Carbon, Alkaline or NiMH. Working voltages then can be as high as 3x1.6 = 4.8V to as low as 3x1.0 = 3V.
This voltage range (3.0 to 4.8V) is well covered by 1x protected Lithium cell (2.8~3.0V to 4.2V).
So, about voltage range - YES, you can replace the battery source to one lithium cell, without major modifications, apart from the Charger itself, as you linked.
About reliability of the Lithium cell and charger:
As you somehow mentioned, several more affordable consumer items are purchased with less than stellar reviews and traceability, but still “work OK”. As most TP4056 modules and lithium cell work acceptably, I would give a try and do this “upgrade” and monitor the start-end charging and discharging/cutoff of the lithium+TP4056, still outside the headphone. Please also observe the IC controlling the Li-Po cell:
Moreover, the battery itself may have its own battery management IC, as highlighted in above picture and vendor’s post:
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Because of that built in protection the use of an already protected Lithium cell + TP4056 may be redundant and eventually cause interference.
Prototype Testing and Verifications:
Some further investigation may be good: , using these 2 parts are necessary and work ok. I would also check for overcurrent: planning a test to use resistive loads to check the said overvoltage, undervoltage and overcurrent works as expected.
As most probably everything will run fine, you may not even need the TP4056, because the built-in protection does all that. Check this video from Kasyan TV about these protection modules.
Final assembly:
After these tests, I would make the final adjustments and overhaulings, as a cabling and charging port connection, etc.
Addendum & further readings:
Although the OP has already accepted the answer, I found some information of certain “TP4056 + DW01A + FS8205 module” (google it), like the one the OP might use.
The following picture (from here) is from a TP4056 with Protection Circuit:
Such module has the following block diagram from here, where it further details and explains the module features:
Additional ”hack” using TP4056 + LiPo battery are worth to see here, here and on another Q&A here.