It is REALLY hard - it's a growth spurt - there is nothing you can do but hold the line and feed, keep feeding, and it WILL end!
I have been where you are, with bf DD. It won't last long, honestly.
WRT the rice - NO. Definitely not, but you know that really! We've all been there with the idiots well meaning people who push baby rice at a ridiculously early age. It WON'T HELP and it almost certainly WILL HARM.
Think about it. Your milk is what he's used to - thick and full of calories. Replace some of that with what is, essentially, powdered nothing. No density, not half the calories. Will he feel fuller for longer and sleep more? Probably not. Will he be better nourished? Absolutely not, far less calories (which would help him grow more and get through the spurt quicker). Will it harm his digestion? Well at this stage his gut is immature, evidence says that YES it can harm in the long term, in the short term it could give him digestion issues which could send his sleep right out of the window anyway! And if you upset his feeding schedule now by encouraging him to take less milk than he needs by filling him with dust - then that's not going to help any long-term routine.
People basically look at a 'solid' and think it's 'proper food', as opposed to milk being a 'drink'.
It's exactly the opposite. You'd feel fuller and happier for longer after a pint of full-fat milk than a huge bag of popcorn - full of air and nothing. That's baby rice.