Rice bran as a single oil soap will be a take a little longer to set up in the mold if made with the default amount of liquid in your
lye calculator. So I suggest you set the lye concentration to maybe 33% for your first batch. If this is your first soap, 33% lye concentration seems like a safe place to start. I'd probably do 40%, but not when I was new.
You can use the rice milk instead of water when you make the lye solution. Just remember to ALWAYS add the caustic soda (Sodium Hydroxide Lye) to your liquid. NEVER add liquid to lye. Add it slowly and stir until completely dissolved. It will heat up. Remember to wear goggles and don't breath in the fumes while stirring, in other words, don't lean over the container. You want to do this in a well-ventilated area. Sorry if you already know this. If you haven't read all the safety precautions on working with lye and soapmaking, please make sure that you do before making the soap. If you have, great!
I suggest using
Soapee.com or
SoapCalc as your
lye calculator. Both are fairly easy to use.
Here is a
link to a single oil soap experiment done with a few different oils. The one for Rice Bran Oil soap is at the end of the page. It got good reviews in terms of lather and hardness. Here is another
link to another such experiment, and the rice bran oil soap is toward the top of the page. I don't think it's a long lasting bar, but that is not an issue for you from what I can see. It's more about something your skin can tolerate.
I empathize with your sensitivities and allergies. I once knew a woman diagnosed with aquagenic urticaria. I cannot even fathom what that is like to live with on a daily basis. So give the rice bran soap a try and I do sincerely hope it helps.
Although you will need to be sure to give a good 6 to 8 week cure before you test it out.