If you want to prepare biomolecule-heterocycle conjugates, you don't have to limit yourself to azide-alkyne cycloaddition, but you can use a variety of methods, including the thiol-ene chemistry or the "standard" carbonyl group reactions, you just have to select an appropriate reaction to match:
1) the structure of your protein and your small molecules
2) the application of the intended bioconjugate
3) the conditions in which you want to perform the reaction (in vivo/in vitro/isolated protein)
For azide-alkyne cycloaddition, the difficult part may be efficient and selective introduction of an azide or alkyne fragment into your biomolecule.
For the cycloaddition reaction itself there is a number of efficient protocols available, from which you can choose and switch/modify if one of them fails.
If you go for copper-catalyzed azide-alkyne cycloaddition, you may have a look at the protocols that we have developed for small organic molecules using a new water-soluble ligand, AMTC.