The extra 'computer noise' you're describing is also caused by incorrect grounding. I experienced this exact problem with my USB DAC. Fixing the grounding solved it completely. Also, I recognize that you can influence the hum by touching the case. This is also something I experienced with my Stax amp, and which I solved by fixing the grounding.
Probably, if you were to ground the case directly with a copper wire to earth, your problem would be solved. This helps for me. Unfortunately, the SRM-Xh doesn't have a ground binding post on the back, like the bigger Stax amps (for precisely this issue!), so you would have to improvise here. It would be a shame to scratch some paint of to attach a wire somehow. What you could try, is to ground the source in your system, if it has a screw somewhere on the case under which you could attach a copper wire to earth. I used to use the bindings post on the back of my Stax SRM-717 to ground my whole system, but now I do it via my computer case. It doesn't have to be the source, it just needs to be the next (or 'next to next') case in your system that is connected to the SRM-Xh. The RCA cables from the source to the amp connect both of their cases and hence their ground (unless your source is double insulated), so if you ground the next case, you'll ground the SRM-Xh along with it. Basically this is what you do when you touch the case: you are providing a path for the electricity on the case to escape the system. You must provide this path in a different, permanent, low-resistance way to solve your problem.
How easy this is, depends what kind of outlets they have where you live, and in some cases how old your house is. If you do not have grounded outlets, you can try grounding your system by tying the copper wire to a radiator, water pipe or the ground of the radio/television cable. This is not exactly how it's supposed to be, but it works well in a lot of cases.
The reason it works normally with your other sources, could be that these have grounded power cords to grounded outlets, while your raspberry pi and Stax amp both do not have grounded cords to their cases, leaving you with an ungrounded system.
In my opinion, a Stax SR-207 is VASTLY better than a Sennheiser HD650, for any genre, but especially for classical music. I owned the HD650 for a long time (back when it was considered high-end), and have extensive experience with a 207. In fact, I have one here right now since a few months, and as I'm writing this I'm listening to some Bach cantatas.
If you want open and airy, you must pick the 207. There's no competition between the two, the 207 is miles ahead in that regard. Because classical music is often recorded in large halls and churches, hearing this spaciousness adds greatly to the music. It sounds so real, so spacious, it's marvellous, even better (also different) in that regard than my Stax SR-007 you could say. On top of that, the 207 has so much more resolution and speed that every instrument is rendered in a completely effortless way, even when the music gets very full of all kinds of instruments, percussion, choir etc. It's easier to make out every little detail, and the music just seems to exist, rather than be reproduced by a driver. You'll hear so much more information with the SR-207 than with the HD650, and especially with classical the experience is (therefore) so much more pleasing with the 207. Classical recordings are often very well done, making it rewarding to listen to them with an electrostat.
I'm not usually this strong voiced when it comes to comparing headphones or other equipment, but in this case I really feel that the 207 is many times better than the HD650. In general I find electrostatics to sound clearer than dynamics, and this is another aspect in which two headphones couldn't be further apart: the 207 super clear on the one hand, and the HD650 veiled (YES) on the other hand.
The HD650 does have a lovely midrange and a (more than slightly) dark signature. However, this soup it pours over everything can't be switched off. Yes, it makes for a cosy, warm sound with classical and vocals. It's more like an old opera house with heavy textiles on the walls and all over the place, dampening the higher frequencies, while the 207 is more like a modern concert hall or church with harder materials, letting you hear all the overtones as if you were close to the instrument.
The 207 is by no means bright. I have heard bright headphones (Stax SR-507 and Sennheiser HE60 for example, also HD800), and the 207 is not that. It's a bit mid-centric, but not by much, it is probably the most neutral pair of headphones I've ever heard, along with the 202. I also prefer the 207 and 202 to the HD800 by the way. The 207 does not have any irritating peaks in its frequency response. The term 'neutral' often gets misused by people who are trying to trivialise the bright nature of a pair of headphones, but in this case it's true: the 207 is not bright.
It is my strong opinion that you should choose the 207, and accept the fact that it will not have the dark signature you search. You could try to attain this in your system by choosing a dark sounding DAC, if you must, but try the 207 first and see if you still want it dark.
Combining an open and airy sound with a dark signature is very difficult to do, because more high frequency information leads to a better defined rendition of spacial cues. Thus, a bright headphone will sound more spacious than a dark headphone, and in a sense, you're asking for the impossible. The SR-007 however, almost pulls off this difficult combination, meaning that it does have slightly attenuated highs and smaller soundstage, but also a very precise placement of instruments and an incredible level of detail, and on top of that a lovely midrange. It's in a different price class, obviously, but within the Stax range, I guess it's the most HD650-like.
