Just a little update on this: I found someone with an SDS2204X+ at work who could reliably reproduce the issue on his scope. Consequently, I created an error description, including steps to reproduce the issue and equipped my dealer with this, to contact Siglent about it. Hopefully, they can locate and fix the issue. Until then, I have to fall back to only using the left half of recorded data.
After months and several attempts of not following my step-by-step tutorial for reproducing the issue, Siglent was finally able to confirm and solve the issue. Apparently it was not a firmware, but a hardware issue, since they replaced the whole mainboard after they could not pinpoint the exact source of the issue. After that, I was not able to reproduce the issue any more.
However, upon in-depth testing of the repaired scope I discovered that the logic probe of the scope is faulty. The signal path of channel D0 seems to be broken. Probing the same signal with D0 and D1, it is only displayed on channel D1. This is not an issue of the probe head, since swapping in another probe head does not improve the behavior. It is, however, possible to capacitively couple a signal into D0 by putting the same signal into D1 to D7.