What exactly are the legal problems with traveling to this region?
Georgia considers Abkhazia Georgian, so when entering Abkhazia from Russia, because of not being checked by Georgian authorities, in their eyes you're entering Georgia illegally.
So if you're stopped by the Georgians at the Abkhaz-Georgian border (which may not happen) and they see you entered Abkhazia from Russia, you'll be heavily fined (400 lari last time I checked - basically 150 euros).
Even if you aren't checked, I recently discovered the Georgians systematically records entries and exits of foreigners (when exiting by bus for Turkey, they even knew I had entered by air from Italy), so if there's no entry record, they may well suspect you of having entered Abkhazia from Russia, or at least entered Georgia illegally. Thus, even if you use your ID card (thus no stamps), you could well be in trouble.
Furthermore, you'll be refused entry to Georgia in the future if presenting a passport with an Abkhaz stamp from Psou. However, if using your German ID card and not your passport at Georgian border control in the future, they obviously won't see you crossed the Russian-Abkhaz border (but see the previous paragraph)
If this direction isn't possible, how about the other way around (traveling from Georgia to Abkhazia through Zugdidi and leaving Georgia/Abkhazia at the Psou/Adler border crossing)?
Georgia always records your entries and exits through its official border crossings, so if you're unlucky they'll instantly suspect you exited Georgia illegally. Anecdotal evidence, however, suggests it's rarely a problem in practice in this direction
In any case, to be on the safe side, go to Georgia through the ordinary crossing point at Verkhniy Lars, then to Abkhazia through Zugdidi and then back to Russia the same way (i.e. not crossing at Psou)
If you must cross at Psou, do so going from Abkhazia to Russia and not the other way round, and if your passport is stamped either by the Abkhaz or the Russians at Psou, use your ID card for any further travel to Georgia