lut5 multiple functions on same inputs

Thanks. I never used select8 and weighted_sum, so need to understand them first to understand what You wrote

The weighted_sum is used to input an array of bits and convert that to an s32 number.
linuxcnc.org/docs/devel/html/man/man9/weighted_sum.9.html
The select 8 takes an s32 value from 0-7 and sets one of 8 bits true.
linuxcnc.org/docs/devel/html/man/man9/select8.9.html

How many gray-code bits are there? do you need bit-mode outputs or something more interesting?

For 4 bits of gray-code and a button you could put the button onto one of the bits.
For more bits of gray-code it is more difficult.

The various mux components are another option.
linuxcnc.org/docs/2.7/html/man/man9/mux_generic.9.html
You can set one of those up with as many input bits as you want, though you then need a "setp" for every possibly input combination. But the output value can be int or float, rather than just bit.