You can't DO that!
I'm sure you're aware that sometimes a junction box can be a bit in-set from the wall - and that's OK if a) the wall material is non-flammable (drywall, brick) and b) the inset is less than 1/4". Yeah, that's allowed, see NEC 312.3.
That does not work with panels. Panels are not junction boxes. On panels, the cover must fit absolutely flush, with zero inset. Read the installation instructions for the panel, which you must follow per NEC 110.3(B).
Why? Because panels rely on proper fitment of their covers to hold the breakers in. Note how the panel's edge catching the breaker is what keeps the breaker from tipping out when you throw it to the "off" position (e.g. urgently). The last thing you want is the breaker coming clean out, which is what would happen if you mounted a panel cover 1-1/4" proud.
I have a panel designed for offset, though
Except the 1/4" limit on inset (312.3) still applies. Remember, all equipment must be built to the UL White Book standards and approved by UL - and UL's requirement harmonizes with NEC. Thus, the panel was designed with that NEC 312.3 limit in mind, and is not designed nor UL listed to exceed it.
Further, you are required to use UL approved equipment (NEC 110.2) which means not modifying it in non-approved ways... and required to follow instructions and labeling on the equipment, see NEC 110.3, which means using the feature as intended. I wonder if the oddball screws are a UL requirement to keep people from doing what you're trying to do :)
How to fix it
You will need to re-work the wall treatment in the immediate vicinity of the panel to remove ALL wall material which is proud of the panel face. How you trim that is up to you, but it has to happen. Don't forget that nice wide lip on a flush-mount panel cover is there to help the drywaller conceal the drywall edge. If you would rather have a surface-mount cover (with no lip) for finish reasons, talk to your Siemens authorized dealer.
Note that if you are relying on the 1/4" rule (NEC 312.3) to have wall material slightly proud of the panel cover, you must also comply with 312.4 and 'mud up' to within 1/8" of the panel sides. That is to keep flame from sneaking through the gaps.
Your best bet, especially in light of your soundproofing goals, is (after you have done the above) to cover the whole thing with a framed cabinet door of some kind. 1x4's around the perimeter then an appropriate cabinet door, possibly with soundproofing on the back of the door, taking advantage of the void behind the door.
Junction boxes, too.
Also revisit all your junction boxes and make sure they are a) within 1/4" of wall surface and b) the surrounding wall surface is a certified non-flammable type (which unfortunately a lot of synthetic stone is not). If not, the cure is not too bad - you can either install an extension box or a telescoping extension sleeve. Such extension products are impossible for panels because the cover must have a specific spacing from the the breaker bus assembly - well, your particular panel has a telescoping spacer built into the cover, but it is only designed to telescope 1/4" or so. Much beyond that, it will not protect.
Lastly, note that all panels require a working space the width of the panel (at least 30"; not necessarily centered), 36" standback distance from the face of the panel (that critical dimension we've been talking about), and 78" (6'6") tall. (and not on irregular ground or stairway). This area must be kept clear at all times. Your AHJ might argue that includes "wall materials" but they'll probably let an inch or two of wall material, or even a cabinet door, slide if everything else is in order.
And by the way, I see a lot of Romex sheath inside the panel. Don't ever let anyone tell you that's a code violation, it's not. It's rather handy for keeping wires grouped, neatly consuming excess length, and identifying circuits.