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

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

• Device A launches application a, device B launches application b.
• The app uses GPS and bluetooth central backgrounds.
• App A fades into the background on device A.
• Application b runs in the foreground and requests a file from device A.
• Device A responds to a file request while application a is in the background and the user has no input to device A.

My first attempt was with Multipeer Connectivity, until I heard it couldn't be done in the background. Stackoverflow question

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

Assuming internet access is not possible, is there anything else for me before trying to fix this by working with BLE and compressing my data?