Yes, with more duckweed and more patience I might be able to run both tanks without filter and still get the ammonia/ammonium and nitrite levels under control. So, I did cheat a bit by adding a filter, but I just wanted to speed up the process a bit.
I did some more water tests yesterday.Now both tanks are like this:
NH4 = <0,05
NO2 = >1
NO3 = 80
pH = 7,5-8,0
I also did a 50% water change and poked the soil with a clean bbq stick. The interesting part was that tank1 (with tiger soil) had a lot of air pockets in the soil. Every time I poked I got air bubbles, in contrast, I got zero air bubbles when I did the same with tank 2 (sowing soil). Both tanks were set up two weeks ago.