Jack,
First of all, I think you are getting your theory mixed up when you say 3-11-b5-5. You cannot have an 11 on any scale that has a maj 3. Nor a b5 and a 5 in the same scale. A proper analysis would be either 3-4-ch-5 or the DCH perspective which is 3-ch-ch-5.
In your fragment the 5 is being approached from below using a double chromatic approach. Yes, the 4 happens to be diatonic. However, in the context of DCH for the sake of simplicity in our thinking both when applying it and analyzing it on paper, we view it as 2 consecutive half steps starting a whole step below the target chord tone.
As far as the name “double chromatic” goes, it is usually going to consist of a diatonic and non diatonic half step. So in this context it is using the term “chromatic” not strictly in the sense that it is non-diatonic, but in the sense that it resolves “chromatically” by 2 consecutive half steps. The need for this perspective will become clearer when you get into 3 and 4 note enclosures.