A difference I see between your recipe and other alternatives is that you don't have any other starch besides your rice. Other recipes include a good amount of tapioca or potato starch. The other difference was that other recipes used rice flour instead of rice.
You might also want to consider a change in technique. You can make rice noodles like a crepe rather than in an oiled steamer pan. Simply use a non-stick or oiled crepe pan, put in batter to crepe thickness, and let it cook until it begins to pull away from the edge of the pan. Invert and release in one motion onto a cutting board and slice.
I hate to say it, but I'd be willing to bet that gluten-free udon is about as practical as gluten-free seitan - the gluten is precisely what gives udon noodles the texture that makes them so special.
Having said that, I've bought frozen udon noodles that had tapioca starch as an ingredient in addition to wheat flour, and those were some of the best udon noodles I've eaten.
The other catch is that making udon noodles at home is hard - traditional recipes call for putting the very tough, springy dough in a plastic bag and stomping on it for a few minutes, as the only practical way of kneading it.
This site describes how to make udon noodles with tapioca - you could try replacing its flour with the gluten-free blend described here. I would also consider adding xanthan gum. If you're feeling adventurous, This e-book (not free, but looks pretty solid) discusses a wide variety of gluten-free flours and their properties, and might help you choose flours to use with the properties you want.
If gluten-free udon turns out to be impractical, I recommend soba noodles as an alternative. Those can much more easily be made gluten-free (although most commercially made ones available in the US do have wheat simply because it's cheap and bland) and can usually be used in the same dishes as udon. They have their own distinct texture, very different from udon, and a deliciously hearty flavor.
Best Answer
Homemade noodles have a completely different texture from dried pasta. The times I've made lasagne for friends with fresh spinach noodles, they were quite confused by the texture. And even if there's lots of moisture and time for it to cook in the casserole, you need to boil them for 30 to 60 seconds or they'll disintegrate in the casserole.
Depending on the recipe, I might consider starting with everything already hot, the pasta boiled for a minute or so, assemble it, then put it under the broiler (grill, whatever you call 'top heat only' in your country) to get a bit of a crust on top, then let it sit for a few minutes to firm up some.