I get what you’re trying to achieve but I feel that induction charging is inefficient for this application.
For this to work, you have to go battery (DC) - inversion (go to AC) - induction (wireless) - rectification (AC back to DC) - decoupling and regulation - charge the earbud battery. Each of those processes has an energy loss that quickly mounts up.
Wireless charging is really only effective if your original power supply is AC. With a battery supply, you may as well design some sort of connection so the process can become: external battery - connector - charge earbud battery. The concepts you already have could be easily adapted to facilitate this.