Well, seems like I did the right thing to buy the 12Mhz-Version instead of paying 20€ more for the 25Mhz-Version
What I discovered when I powered the unit from my Bench-Supply (Ebay-Seller had sent me a PSU with the wrong type of Plug for European Wall-Plugs) was that the waveforms looked a lot less noisy than when powered with the PSU delivered with the unit. I also measured only 4.75V getting into the unit, suggesting the PSU is not really up to the task.
When I powered the signal-gen with the Bench-Supply, I found a significant improvement on the overall performance when I turned the voltage up to 5.30V (I didn't go higher, as I suspected the 5V-Rail to be directly connected to the PSU): The Display had more contrast and the waveforms didn't seem to flatten at max amplitude.
Also a slight offset of -1 to -2% yielded a much better looking result on the Square-Wave.
Unfortunately, I haven't made any comparison-pictures
I am currently investigating upgrading the unit with a battery and an internal switching PSU: The one I got with the unit clearly is not up to the task, but I'm pretty sure one of these cheap chinese Boost-Converter Boards + a 3,7V Lithium-Accu combined with a small charging-circuit connected to the original PSU-Input, will be. And there is more than enough space inside the case to add this.
I also think about modifying the on-board switching regulators to put out +/-15V or +/-12V if the caps are rated for only 16V. If there is nothing else powered from these rails than the AD812, I think this should be possible without risking damage from overvoltage.
The current-draw at 12Mhz, 15Vpp into 50Ohms was about 450mA @ 5,3V btw.