Pull up your device in MB and check the settings tab for 2-way synchronization. I may be mistaken, but I seem to remember that feature being able to perform the task you desire (and I'm also pretty sure that it was supported by my Moto Z with the SD card mounted as an external storage device).
Alternatively, you could look into indirect/2-stage syncing over wifi:https://musicbee.fandom.com/wiki/Synchronize_Android_by_Wifi
Theoretically, you could permit both clients to modify the files and apply the most recent changes to all devices.
As for Android apps, all I can say is that I settled on Pulsar but it won't
work for you because it uses the "love it" or "neutral" rating seemingly independent of the files' metadata. I'm ok with that because I only sync 4+ star files and very rarely change them.
I too would love to hear if there is a recommended app since there is [very sadly] no MB for mobile.
And I don't know what is the Fiio DAP of which you speak, but I've seen that some dedicated DAPs run a version of Android--and if it does, perhaps either of the methods above would apply to that too.