Indeed it isn't possible. Consider any uniformly continuous image of those irrationals in the Cantor set. The inverse image $X$ of a clopen set, i.e. a finite union of the smaller "copies" of the Cantor set in itself, is going to be "uniformly open" (there exists a $\delta>0$ such that every ball of radius $\delta$ with a center in $X$ is entirely contained in $X$) and closed.

Such sets in $[0,1]\setminus\mathbb Q$ are either empty or the full set, by the usual proof that $[0,1]$ is connected. In particular, $y\neq z$ in the Cantor set can never both be values for the map, since the preimages of two disjoint clopen sets cannot both be full. Hence the map is constant.

**Edit.** This answer is basically an unfolding of [the argument of bof](https://mathoverflow.net/questions/463516/is-there-a-uniformly-continuous-injective-image-of-0-1-setminus-bbb-q-in-the/463517#comment1203433_463516) in the comments (existence of an extension to $[0,1]$ corresponds to the fact that the preimage is "uniformly open").