I know I didn't make it clear in my description: but I don't imagine that we will want (or be able) to send strings of digital information to the fox transceivers in the spaces between the code elements. Instead, I was thinking that we need some way (any way) to flag the fox transceivers to stop transmitting and listen for commands. I am quite certain that each fox's receiver will be totally desensitized (deaf) during its own transmissions, and unable to receive any in-band signals while its own transmitter is putting out a signal.
There is also a chance that some foxes will be close enough together that they are deafened by the transmissions of nearby foxes.
If several, or in the worst case all, the foxes' receivers are desensitized by the other foxes' transmissions, then it could be very difficult or impossible to send any messages to them while any of the foxes is on the air.
But if the foxes are capable of hearing signals between code elements (QSK), then you can guarantee that all foxes will be able to receive a sufficiently-long signal burst - a signal burst longer than the longest code element that any fox sends. Even the fox that is currently on the air should be able to detect such a signal (filling in the space between its transmissions) and shut itself off in response.
So with QSK, if you want to get all the foxes' attention, just send a 10-second long (for instance) transmission. In response, all the foxes will go quiet and start listening for commands, even the fox that was actively transmitting. Then the control station (it could be on the homing beacon frequency, or any other frequency all the foxes' receivers are tuned to) would be able to send digital messages to the foxes, and know that they are all listening. The messages could be very slow, and repeated several times, but with no fox transmissions to interfere.
But QSK isn't totally necessary. There are other schemes, and the one you are proposing might be the way to go.