Stubborn CuAAC reaction as derivatization for ESI-MS
How to get a better turnover while maintaining a reasonably short reaction time?

To speed up the reaction, you should look at changing (1) the reagents, (2) the catalysts, or (3) the reaction conditions.  In particular you might be dealing with poor solubility for your alkyne substrate if you are doing the reaction in aqueous conditions.
1. Reagents
You could try the picolyl azides I mentioned in the comments in lieu of the 4-acetophenyl azide you originally tried.  These azide reagents chelate copper significantly, thus increasing reaction rates since proximity of two of the three components of the reaction (\ce{Cu+} and aizde) is enhanced by the chelation effect.
2. Catalyst
There are a variety of exotic $\ce{Cu+}$ organometallic complexes that work better than the &quot;standard&quot; copper salts.  An example publication is Sebest et al. 2018, which reports that the N-heterocyclic carbene complex $\ce{[CuI(Mes‐6)]}$ (Mes-6 being an NHC ligand pictured below) worked well at loadings down to 0.05 mol% in their trial reaction.

This might be overkill if you are not using the ligand THPTA used in your original protocol.  Maybe you could just use that?
3. Conditions
You an always heat more, but I think a more important variable is your solvent.  Is your reaction solvent one that dissolves all the species, including alkyne, azide, and copper(I) complex?  Instead of aqueous conditions, can you try acetonitrile or 100% DMF?  Or n-butanol?
Additionally, the Presolski reference you give mentions that high chloride concentrations (~0.2 mol/L) can compete with ligand for copper ion.  Plasma has about a 0.1 mol/L chloride concentration, so maybe it's an issue?  Probably speaks to the need to use a good copper ligand like THPTA.
Bottom line
If it were me, I'd try getting some THPTA or other ligand to improve $\ce{Cu+}$ solubility in organic solvents, and trying the reaction in non-aqueous solvents where your steroid is likely to be soluble.  I'd also get the picolyl azide if my lab could afford it.  If that doesn't work, and I still wanted to improve things, I guess I'd look at increasing temperature or using more exotic NHC ligands for the copper.