If you do get a new battery, I would either ask apple to see how much they charge (I think it’s pretty cheap) OR get one from ifixit. They have pretty good batteries. If you go the cheap route, you will get what you pay for believe me.
Obviously, you probably have given up on this or never thought about it again but I thought I would chime in. So, it could be the battery or I would check the charging port. Make sure nothing is stuck in there. Also check the audio jack, that could interfere with charging. Make sure you have the correct charger as well. If none of the above work, I bid you farewell.
Before you embark on soldering a Bluetooth chip onto the thing I would recommend something like this: https://www.amazon.com/KOKKIA-Transmitter-authentication-capabilities-generation/dp/B004SH40BQ/ref=pd_day0_hl_...-8859559-5906731?_encoding=UTF8&pd_rd_i=B004SH40B...-3481-11e9-ba4c-998c19a870f7&pd_rd_w=8fGG2&pd_rd_wg=Viz4...-e646-4161-82c7-5ed0d4c85b07&pf_rd_r=JMM9GCKEA2M9X01XMER... If you still want to get crazy you can try this: https://www.instructables.com/id/Bluetoo...-IPod-Classic/ Good luck sir!