There was a recent paper on labeling tyrosine via Copper free Click chemistry: You could try this protocol with our dye DBCO derivatives.

An alternative also makes use of tyrosinase, but with hydrazide treatment instead: The paper contains an example with Cyanine5 hydrazide. We have a selection of dye hydrazides.

A purely chemical approach makes use of diazonium salts to modify Tyr, but the labeled derivatives should be further modified with reactive dyes. I could imagine a protocol based on alkyne-modified diazonium that could be further labeled with dye azide, but we do not have all reagents for this chemistry (you will need to source alkyne-modified aniline somewhere, and generate an unstable diazonium salt using sodium nitrite and acid right prior to the reaction). However, it is possible to make use of enzymatic labeling described in the papers above, this might be an easier and milder method.