Don't know for sure since I don't use Reaper, but I noticed something interesting.
I create a track to contain Mobius, insert it as an effect. In order to get MIDI to flow from the reaper MIDI inputs into Mobius I have to enable that track for Recording. You can now send Notes into Mobius and it responds to them.
But when you start the transport, if you have enabled recording in the transport, it's actually going to record all those Mobius command notes into the track. You never want to record the MIDI that is being sent in to the Mobius track, but if you don't arm it for recording, it won't receive any MIDI.
Since Reaper thinks it is doing a normal recording of keyboard notes into the track, it may be trying to be smart about stuck notes when the transport starts and stops, even though these are just plugin command notes, not anything you would want to capture and save.
Is there a way to get MIDI into a track and its plugins without arming it for Record? Maybe some kind of Monitoring option?
Ableton and AFAIK Logic doesn't seem to have this issue because they distinguish between audio tracks and midi tracks. Mobius normally goes in an audio track since it is a sound generator, and receives MIDI only for control. But these two concepts seem to be merged in Reaper.
[update]
I found a menu where you can select "Record: disable input monitoring only".
Also in here it looks like you can configure the track to be only MIDI or only Audio, not "audio or midi".
So you might try using that "input monitoring only" option.
Also too, over in the MIDI Captain, it is important that the switches be configured as "momentary" where they send different events on press and release. If they're only sending NoteOn, then this may be why Reaper wants to get involved and turn notes off. This actually wouldn't work at all for the Mobius Record function since without a NoteOff when the switch is released it does a "long press record" which will Reset the loop. But if you're always using AutoRecord, you wouldn't notice this since AutoRecord doesn't have long press behavior.
Hardware controllers must always send NoteOn when pressed and NoteOff (or NoteOn velocity 0) when released, or CC > 0 when pressed and CC 0 when released.