Modification of Speaker-System to operate at 12VDC (Logitech Z903)
So after a long break I returned to the project and I was able to solve this.
It turned out that the Pin  “+3.3V” of the “Mainboard” had been tied to HIGH (3.3V) by the Power-Supply PCB.
Thanks to @RonBeyer I got the idea of testing it this way.
MY SOLUTION:
As mentioned in the question above I replaced the Power-Supply-PCB by the following:
To get the 45.5VDC from around 12VDC (car battery):

1200W 20A DC Converter Boost Constant Step-Up Power Module 10-60V to 12-83V (just a no-name product being sold everywhere e.g.: eBay)

As the loudspeakers main-board also requires 12VDC and I was not sure on how it would react to a fully charged 12V battery (max 14.2V) I added a step-down buck-converter like this:

Buck Converter Pack from Amazon

For the 5VDC supply I also used the buck-converter from above.
In order to tie the mentioned 3.3V Pin to HIGH I used a very small/cheap step-up/down module from Amazon (for the input I used 5VDC from the bucket converter above)

Step-up/down module from Amazon

Additionally I added a relay for switching on/off the power-supply.
I hope this helps someone.