My assumption (and yours, I guess) is that if the movers dont have DMX512 data coming their way they "fallback" to sound to light mode. If that is the case, then what you are trying to do
should work, so the devil is, as ever, in the detail.
You have read the rants about DMX512 termination, haven't you - 'cos my money says thats where it's all coming unglued. You have got a terminator at the end of the DMX512 run haven't you.....? I thought not
The relay: use a DPDT relay. Wire the commons of the two poles to the movers A and B inputs, the normally open contacts to NSI A and B outputs, which is probably exactly what you've already done. Now wire a 120R resistor across the normally closed contacts. That provides the bus termination when you dont have a desk connected. Energise the relay to give control to the NSI interface.
That will work, and when the relay is open the DMX512 bus will be silent. Note that everything will probably "glitch" as you connect and disconnect through the relay,as the switching is not syncronous with the data. This shouldn't be a problem of any magnitude, but if it is you need a fancier (and thus more expensive) solution.
Some movers with sound to light and other self-running modes are cleverer than others, and the DMX512 bus isn't actually a daisy cahin, althopugh it looks like and is wired that way, but a point to point link between each device, and the first one in the chain can generate control info for the rest, so they behave as a single collection of fixtures, rather than a bunch of fixtures doing their own thing.
Let us know how you get on!