I made this repair today without too much difficulty. I watched the following video to know what to expect. I kept the speaker attached to the cup which made the repair a little awkward to navigate, but nothing major.
https://www.youtube.com/watch?v=Ju4J-ZwKsb8
Gentle work with a plastic spudger will remove the right cup with a series of loud pops, then 8 screws to open the Power Button (right) side of the head phones. I used two jewelers' screwdrivers to remove the power connector of the original battery, which is held in place with generic VHB double-sided black tape underneath. Mine had begun to swell so this was a delicate operation with soft tools so as not to puncture anything.
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As noted in the video and above by Israel Lopez, make sure the polarity of your battery connector matches the replacement. Mine did not, and I had to very gingerly remove the red and black pins, bend their retaining clip up a little (doesn't take much) and reinsert them into the JST connector to match the original. I did not find a need to remove the headphone PCB.
At this point I connected the battery before securing it down to make certain the headphones powered on. When they did, I secured the new battery with double-sided tape, placed a small, inside-out roll of painters tape on the top to sit between the speaker magnet and the battery itself, and reinstalled the cover with its eight screws, then the ear pad with another series of loud pops.
jayeffs battery information above is phenomenal, and with it I found this ~$7.00 1800mAH battery that fit perfectly with no modification necessary, and is a significant capacity upgrade.
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https://www.amazon.com/dp/B08Y6R2GMQ
"EBL Replacement Sp 803048 Battery 1800mAh for Infant Optics DXR-8 Video Baby Monitors, High Capacity 3.7V Lithium Ion DXR-8 Rechargeable Battery"