If the sound is clean between the gaps, it is probable that either the sampling rate being delivered by the TV is not close enough to 48kHz for the DAC to achieve a good lock or the digital output of the TV is itself a bit twitchy. It is far more likely to be a TV issue than the DAC, assuming the DAC is working as it ought when fed from other sources. Does the DAC remain locked and stable during the gaps or does it go off-lock, recover and the sound returns?
I assume that the black Samsung unit in your picture is some sort of breakout box from the TV? Does the thick white wire connect this box to the TV? Does the TV not have an optical socket to which you can connect your DAC directly (without using the breakout box)? Check that the connectors to the breakout box are properly mated at both ends.
Many TVs only reproduce as good (or as bad) a digital audio signal as they are given – GIGO – Garbage In, Garbage Out. In other words, if the audio coming from your satellite RX is twitchy, the TV will not do anything to improve things. If the audio is stable when playing sources other than satellite, I think that tells you that the audio coming out of the satellite RX is poor and/or the TV is not re-processing digital audio effectively. This is a design issue with many TVs and may not be easily solved. In general, the best source to test the TV's audio stability is terrestrial – which is the TV all by itself with no reliance on any external equipment. If this audio is stable through the DAC, it tells you that you need to look at the stability of the source audio.
Additional edit
Can you connect the digital audio output of the satellite box directly into your DAC (bypassing the TV)? Such a test will demonstrate the reason for the problem.