Transferring a file from an iOS app running in the background on one device to another device with an iOS app running in the foreground

Question

I would like to transfer a file between two devices in the following scenario:

1. Device A runs app a, device B runs app b.
2. App a uses GPS, and bluetooth central background modes.
3. App a goes to the background on device A.
4. App b runs in the foreground and requests a file from Device A.
5. Device A responds to the file request whilst app a is in the background and the user has no input into device A.

My first attempt was with Multipeer Connectivity until I had heard that it was not possible to run this in the background. https://stackoverflow.com/a/25517507/1846108

Next I tried with BLE but the max throughput was too low (~ 3kB/s).

Assuming that internet cannot be accessed, is there anything else left for me to try before I attempt fixing this by working with BLE and compressing my data?