Just to update on this, did some more testing:
Using the same dongle in all tests.
From my Windows 10 box, testing via MIDI-OX:
- To DR-3 - no issues noticed. PC, Song Select, MIDI clock, song stop all work
- To other device - no issues. Only tested PC
From Android laptop, using Repertoire
- To DR-3 - PC messages work! Song Select and Song Stop messages don't! Repertoire's MIDI log shows messages are generated, and the outgoing light on the dongle blinks, seemingly confirming this. But only PC has an effect on the DR-3
- To other device - PC messages work.
From Android, using MIDI Commander
- To DR-3 - PC messages work. But can't test Song Select or Song Stop.
- To other device - PC messages work.
On a lark, I switched to another dongle and repeated the tests - got exactly the same results.
My conclusion would be that somehow, Repertoire is not formatting the Song Select and Song Stop messages properly, even though the log seems to indicate the correct behavior. Likely that conclusion is wrong, as you would have heard about issues with Song Select long before now.
Clearly there is a message being sent, but it is just not being understood for what was intended. Unfortunately, I don't have a way of logging the message as it appears on the out cable of the dongle, which would probably help immensely in understanding :-( The only thing I can think of is that PC messages are to specific channels, whereas Song Select and Song Stop do not have channel information attached.

?
Here's the Repertoire Log:
- found RTP port BandHelper - TravelMate 8210
- adding port BandHelper - TravelMate 8210 for input
- adding port BandHelper - TravelMate 8210 for output
- found USB port USB2.0-MIDI Port 1
- adding port USB2.0-MIDI Port 1 for input
- adding port BandHelper - TravelMate 8210 for output
- found USB port USB2.0-MIDI Port 1
- adding port USB2.0-MIDI Port 1 for input
- adding port USB2.0-MIDI Port 1 for output
- found USB port USB2.0-MIDI Port 2
- adding port USB2.0-MIDI Port 2 for input
- adding port USB2.0-MIDI Port 2 for output
- adding port USB2.0-MIDI Port 1 for output
- found USB port USB2.0-MIDI Port 2
- adding port USB2.0-MIDI Port 2 for input
- adding port USB2.0-MIDI Port 2 for output
- sending data [F3, 02] to all ports
- sending data [C3, 01, C1, 02] to all ports
- sending data [F3, 03] to all ports
- sending data [C3, 02, C1, 03] to all ports
- sending data [F3, 01] to all ports
- sending data [C3, 00, C0, 01] to all ports
- sending data [F3, 02] to all ports
- sending data [C3, 01, C1, 02] to all ports
- sending data [F3, 03] to all ports
- sending data [C3, 02, C1, 03] to all ports
- sending data [F3, 02] to all ports
- sending data [C3, 01, C1, 02] to all ports
- sending data [F3, 01] to all ports
- sending data [C3, 00, C0, 01] to all ports
- sending data [F3, 01] to all ports
- sending data [FC] to all ports
- sending data [F3, 01] to all ports
- sending data [FC] to all ports
- sending data [F3, 02] to all ports
- sending data [C3, 01, C1, 02] to all ports
- sending data [F3, 03] to all ports
- sending data [C3, 02, C1, 03] to all ports
- sending data [F3, 01] to all ports
- sending data [C3, 00, C0, 01] to all ports