I think it would not be a problem if you use NanoDrop for your measurements. But there are better options if using UV-Vis is not a strict rule. Alternatively, Qubit itself doesn't have any issues since their reagents do not get interfered by aromatics.
As I understood from your description, you preferred using UV-Vis method, which uses wavelengths around 100-400 nm (UV-A 315-400 nm, UV-B 280-315 nm, UVC 100-280 nm). ThermoFischer provides two options for UV-band measurement using NanoDrop (Table 1 of NanoDrop Protein Quantification page), A280 and A205.
The usual A280 method is what you are concerned about, since its absorbance behaviour is strictly linked to tryptophan, tyrosine, and some disulfide bonds. This limits A280 only to protein samples with those residues. To avoid this issue, Anthis and Clore has developed a method of using 205 nm UV-band instead of 280 nm with promising results. You can choose the 205 nm-band method in the NanoDrop settings.
However, if I am not limited to measuring in UV-band, I would personally use Pierce 660 nm colorimetry method to measure my samples. This paper by Antharavally et. al. describes its flexibility on several substances (detergents, buffers, etc.) and its concentration which might suit your treatments. It is also available in NanoDrop settings, even with much greater sensitivity than both UV-Vis methods.
Speaking of Qubit, their BR (Broad-Range) Protein Assays are based on reagents reacting to N-terminals of proteins, not aromatic residues. There should not be a problem for any protein strands.