Oh, I 100% agree! I thought that laughing smiley at the end of the paragraph was enough to signify my sarcasm/amusement about this Idea! 
Well, the rainbow that the grating produces, just has to be as wide as your sensor…making the slit-width smaller should reduce spectral overlap…am I missing something? The wavelength dependence and thus the angle of the diffraction doesn’t need an optical system to be imaged,…does it? No I think it doesn’t. It may help but I think it’s not necessary (if I am not overlooking something super simple here).
Mulitmode fibers to the rescue!
https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=3255
Not too expensive. But of course coupling into the fiber is the big hassle because of the narrow acceptance angle. The couplers cost a lot, 180EUR here…
I would at least try to just park the camera at a distance to the grating and see of you can make do without a lens. 