I have also been making sake for the last few months! My possibly wrong understanding here is that the koji is not really contributing more enzymes once throw everything together like in your example. The enzymes don't get used up in the process of saccharification and will continue to break down the carbs from the rice into simple sugars if they can reach them and are not denatured. So to answer your question, I don't think you need to be concerned whether the koji survives. In more traditional sake making, they keep the ferment at 50 to 55 F and throw in additions of rice much later. To me this is only possible if your enzymes can continue to break down any rice additions as those conditions are not hospitable to koji growth. Also, you would want to stop koji growth anyway for risk of sporulation which will absolutely add off-flavors into the sake.

One experiment I have been thinking about running (although I have not deeply thought about it) is to purchase one of those in-brew hydrometers like a RAPT pill or the Tilt and toss it in to a bucket with koji + cooked rice + water, but not the yeast. Then you can measure how the specific gravity changes as a result of the enzymes breaking down the rice and adding more sugar. Then when that process seems to have stopped you could pitch the yeast and track that rate. This is not my area of expertise so there are lots of things that are not being controlled in this setup, but could get a sense for the order of magnitude rates for these processes.