Instead of tilling try cutting the rice
straw short and letting it become a mulch layer and just seed right through that with the barley. This way you can harvest while the rice is at the late dough stage so the soil moisture is still fairly good. You also could lay some
water into the paddy to moisten the soil some if needed.
The beauty of this method is that rice straw will slowly deteriorate into the soil and the barley crop will benefit from the retained moisture, it will sprout up through the rice straw easily.
It the rice roots show to be a problem for your seeder, then a harrow run might break up those
root mats enough for the seeder, you just need to do some test patches to determine that bare minimum needed point.
Here a lot of rice farms burn off the straw instead of bailing that straw for sale, to me that is a waste of profits but at the same time it does contribute
ash. The problem is they then come in and do a four stage disturbance in the name of weed control, killing any soil biota they had from the rice crop.
I love the fact that you are using antique implements, in my
experience they work better and are more durable than most of the newer implements.
Plowing is best for incorporation of "green manures" otherwise it shouldn't really be necessary unless you have extreme rocky soil.
Any time we disturb the soil, there should be enough reasons to justify the disturbance as in getting living amendments into place and bringing minerals (rocks) to the surface and breaking these into smaller rocks.
The ideal is to leave the microorganisms that are the soil intact as much and as long as possible. The modern farm method is more about convenience to the farmer than about building great soil, they are the true "dirt farmers" since all their soil disturbance kills or puts dormant their microorganisms.
In my garden spaces the soil might get disturbed once every three or four years after the initial tilling when I incorporate the needed additions per my soil testing and biological assessment (I have my own research laboratory so I use it for the farm too).
Redhawk
PS if you haven't already, you might want to read my soil threads. and if you desire, you can always send me a PM with your questions.