I think this is now all clear.
The Prime tuners on Windows are supported as OpenCable, using DirectShow. The way we do it, we call for a specific tuner each time we need one because we want to manage the availability of tuners. Each time we test a tuner, we ask for its playing state. If it is "PLAYING', we know that the tuner is in use by some app, MC or third-party. If the state is STOPPED, we can grab and use it.
The trouble is, if someone uses a tuner as DMS, it seems that the DMS server and the OpenCable server are not talking to each other. So if anyone uses Tuner0 as a DMS tuner, and MC asks what its state is as an OpenCable tuner, it still returns STOPPED.
So currently the only way to avoid this conflict is to order your tuners so that Tuner0 is ranked lower than Tuner1 and Tuner2. Most likely, a third-party app using a DMS tuner would just do it in an "automatic way", such as it uses the first available tuner, which usually is Tuner0. By favoring Tuner1 and Tuner2 inside MC, you have the best chance of avoiding the conflict.
Yes, you should be able to run your second client to view a different channel, as long as the above conditions are true, i.e. you put the two tuners that the third-party app is least likely to use on top of the list.