[TheMode]

[0] and [1] seem pretty promising to me, although I believe that the problem come from the fact that these languages do not expose a way to describe a sequence of bits, which may make them not lambda calculus anymore, but probably much more usable (you could then implement a soft version of integer/float operations and a compiler that convert this code into a single instruction, easier as there is no side effect guarantee)

> but I haven't yet come up with a good system of combinators that can represent I/O and foreign functions properly

I do not believe that you should support either.

I/O could simply be the program arguments & the final reduction of the expression. The input being your mouse position and the output being a window shouldn't be the responsibility of your program.

Foreign functions are IMO a bad idea, especially if the goal is for the program to outlive you, you lose the guarantee if these functions depend on the environment. Why couldn't you copy/paste the code into your own project?

[0] - https://esolangs.org/wiki/Binary_lambda_calculus

[1] - https://esolangs.org/wiki/Dependently_Typed_Binary_Lambda_Ca...