Short answer: in order to trigger START/STOP on the DJS-1000 (or any Midi device) your stomp box would have to be capable of sending SYSTEM REAL-TIME MESSAGES (Start/Stop/Continue/Timing Clock/etc.).
in the DJS manual, the Mid Implementation Chart mostly describes the mapping of what are called Channel Voice Messages: Note On, Note Off, Program Change, Control Change, Pitch Bend, etc. I looked at the Midi section in the Stomp XL manual and i didn't see anything about sending Start/Stop commands.
In the Utilities for the DJS-1000 you can turn on/off receiving or sending what are essentially System-Real-time Messages (which you of course know about from enabling midi start/stop/continue in settings). But if an external device cannot send a Start message to the DJS, you's have to consider making the DJS the Master opposed to slaving it to the external device.
it may be possible for you to use some sort of 'translator' software or hardware which could take a message that the Line 6 Stomp XL can send and turn it into a Start message (BomeBox by Bome, Midihub by Blokas,are two examples of hardware translators and there are numerous software translators)
hope that helps