Alright, so here's the verdict:
It could theoretically work. If you're using for instance 1, 2 and 3 on the numpad, you can bind the left mouse button to 1 using third-party software, K1 to 3 and K2 to 2. It registers all keys if you use a 'bouncing' pattern like 1-2-3-2-1-2-3-2-1-2-3..., but only if you make sure that you keep actually lifting your fingers; you can not hold down 1 and 3 at the same time or it won't work. You can stream at very high speeds if you keep it consistent though.
The real problem is that it's a very awkward playstyle, even for someone who mostly alternates (like me). You would seriously have to put in a lot of effort to learn this, and I doubt the payoff would be worth it in the end. You could try going for a hybrid style where you mainly single tap with your index finger on 3 and only switch to three-finger tapping at streams, or maybe alternate between 2 and 3 instead of single tapping... Either way, it would be extremely confusing in the beginning. Also, because you cannot have 1 and 3 held down at the same time it puts a lot of stress on your fingers, maybe even more than double-tapping streams at higher speeds.
Regardless, I'm going to master the three-finger-bouncing playstyle because I have no direction in life. Also because people will then know me as 'that one guy that will never be good at osu! because he refuses to just learn how to stream lol'.