Update:
So the middle was not actually bolted down to the printbed frame as I thought. You can actually push it down and it will ‘spring’ back up just like at the corners. It just needs a lot more force. I could not see a way to adjust the height in the middle other than by adding more tape there so I continued with that.
I was getting some heightmap measurements that would vary a lot, even if I did not change anything to the plate or bed, for repeated measurements at the same temperature. I saw a comment on youtube from someone claiming that with the new sv08 bed all you had to do was loosen the screws, heatsoak at 100% and adjust until flat enough. That did not help me directly but maybe the heatsoaking stabilized the magnetic sticker and my tape a bit.
After that I changed my bedmesh temperature to 0 to be able to change/measure a bit faster. I thought I was getting closer to 0.2 but the measurements were still varying a lot.
Then I noticed the horizontal_move_z: 5 and changed that to the 2mm I was using before, changed the bedmesh temperature back to 50C and got more reliable measurements. I also made sure to run the front fan of the toolhead at 100% and park the head at Z=200 while I was making changes. The eddy coil is supposed to be more accurate at that height but the Sovol Eddy has no temperature compensation so I tried to improve the accuracy this way and it seemed to work well.
So I got to the point where the Z height difference is now at about 0.2mm and was finally able to print a 350x350 first layer for the first time. Not a great first layer but I think I can probably still improve this.
Thanks!