Sollya is probably the best modern tool for doing this. Under the hood, it does a Remez approximation followed by LLL to quantize to floating point. No use of Chebyshev directly.
Sollya can also do rational approximants, which are only faster in some circumstances, and Chebfun does not (as far as I know) account for floating point quantization, which is a big deal if you are trying to be accurate.
Possible misunderstanding: I mean a rational function in the sense of Padé approximation or CF [1], not just representing individual numbers as p/q.
I did not find anything related to this in Sollya [2].
Note: results. The software itself is a bit of a pain to use.