I think you are on the right road as to where your problems lie. This involves a known area and I have advised on it many times as most people seem to be unaware of how the H&K amps actually do what they do.
You must realise that the input stage of the H&K TMs and GMs is designed to be very versatile. It acts as a clean buffer, a clean boost pedal, a mild overdrive pedal and a Tubescreamer type distortion pedal according to the signal you put into it. There is no switching for that, it is unnecessary. That is one of the ways the amp has the range it has but using it needs to be learned. We all know you have the valve stages giving their own characteristic sound which is the basis of the amp. But you also have the Gain control set around an active solid state stage which is designed to add in its own mild overdrive if it is set high. You then have the input stage which can be made to give TS9 type overdrive/distortion characteristics too. These are surrounded by absolutely clean control stages, Tone and Volume, which add nothing to the signal at all.
Do not believe the muso hype, transistor/opamp stages neither add nor take away anything from the sound unless they are badly designed. It is easy to design a SS stage to give 0.01% distortion across its whole range of frequencies and levels. At the 0.01% level in a guitar amp that is absolutely inaudible. The idea that they are adding some sort of "blandness" is an absolute nonsense in every way. Any change in sound comes as a consequence of and can easily be detected as a level of added distortion. If you genuinely can hear it then it is impossible you can't measure it! If there is no detectable distortion, and bear in mind test gear is many many times more sensitive than ears here, there can absolutely be no difference in sound to be heard.
A very versatile chain of distortion mechanisms is a deliberate part of the design. It means you can get a large variety of sounds out of the amp. The mistake people make is in trying to use all of these methods together. Distortion on distortion on distortion on distortion WILL sound dire! You must decide what you want to hear then limit the mechanisms to give what is needed for that to be achieved. If you want output stage saturation then turn up the MV with the Power Soak as low as you need but also keep the input signal lower and the Gain moderate. Likewise if you want a good modern high gain sound turn the Gain high with the MV lower and still with a lower signal at the input. If you want a classic pedal type sound turn the guitar high or use a Clean Boost pedal and adjust the Gain starting from Low until you get the right sound then adjust the MV and Power Soak for level. Pedals added into the signal chain are meant to be a method of fine tuning the sound, not a way of creating it. If you want a sound based on a particular pedal (or pedals) then all you need is a flat characterless amp following them. The amp gives you your sound, the pedals simply tweak it. You don't need to spend £/$250 on a couple of transistors to do that really. Most pedals are simply clones of a couple of types. All back to back diodes do the same job and sound the same. Look up the "Schockley diode equation". Here is a link to a nice display of this concept and how it works in the real world too:
Interactive Diode Curve. You can see that the shape of the curve is the same no matter what the diode is and it is simply a matter of scaling to make them all absolutely the same in use. There are no "magic mojo types", that is another myth. A couple of caps and resistors then shape the sound and that is about 20p/¢ worth of components to get a slightly individual sound.
You will create nothing but muck if you try to use a distortion pedal(s), a high level input forcing the input stage into its TS9 distortion, overdrive the Gain stage into distortion, drive the valves into high gain distortion, then try to compensate with the MV. There are too many distortion mechanisms to sound anything like decent. I would strongly urge you to start with no pedals at all. Yes, this goes against modern thinking where, if you are not using at least two high priced (overpriced that is) pedals before your amp you aren't a "real" guitarist. Keep the guitar volume low and try each of the controls in turn from moderate settings, Gain, MV and PS individually to see their contribution. Next turn them all down to moderate and hit the front end with a full guitar signal to hear any increase in that front end overdrive. Then add in a Boost pedal or a distortion pedal set to lower gain to simply lift the signal level to hear the amp front end gradually kick in its full distortion contribution.
Pedals, Input stage, Gain Control stage, Valve stages, Output Stage, all add their own distortions when made to do so. That's far too much if you try to use them all. Keep the five mechanisms of distortion separate and only use a couple at a time, sparingly. That way you will get the right sound without the confusion. This is a brilliantly designed amp with a variety of ways of getting different sounds out of it. Very few people I find ever get to the best of them by trying to be the "ultimate mojo guitar warrior" and doing everything that "common knowledge" tells them they should be doing. Engineering triumphs over mojo every time.