The Mute button in TotalMix only controls the internal hardware Aux sends to hardware outputs (at least, that is the case on the UFX).  It has no impact in your DAW - the input will always be available to software as an audio input.  So you want to be in your DAW to solve this.

In your DAW, you should only be able to hear that track when:
a) you are in playback, and prevoiusly recorded audio is played to you
b) you ARM the track, in which case you can usually monitor the input while the transport is stopped
c) you are recording, and the ARMed track is set up to monitor live input in the track settings

Most DAWs have controls governing these options, as there are situations when you might not want normal operation (eg, sometimes you will be monitoring separately, and only want the other tracks playing as a cue, and not what you are recording as an overdub.)

Hard to be sure without further info, but it sounds like it is something to do with how you are monitoring inputs in your DAW, so I would play with those settings, and see where you get to. 

You should have two sets of monitoing controls if you are using two mono tracks for recording.  Since these settings are push button options, they should not upset whatever balance you have set in terms of gains and volumes on each input. 

Hope that makes sense.