The question as stated is a bit confusing. If your interest is to observe fluorescence resonance energy transfer between tryptophan in a protein and flavin mononucleotide bound as a prosthetic group to the same protein, then you already have everything you need to do the measurement. You don't have to label anything. Trp is the donor and FMN is the acceptor. You can compare the Trp fluorescence in the presence and absence of FMN to measure the distance between them, for example.

On the other hand, if you want to use both Trp and FMN separately as donors and you are looking for acceptors for each donor, then you have a more complicated problem. No canonical amino acid side chain will acts as an acceptor because they none has any absorbance at the emission wavelengths of the donor. You have to introduce an exogenous acceptor. This could be a ligand of the protein with a suitable acceptor attached. Or, you could specifically label an amino acid residue in the protein. This almost always means a cysteine residue, because it can be labeled specifically by commercially available reagents. Often, people make site-directed mutants to put cysteine residues where they want them for labeling. It may be necessary to mutate some natural cysteine residues to a different amino acid to avoid labeling them.

Lysine can also be labeled specifically by commercially available reagents, but there are usually several lysine residues on the protein surface and it is not usually possible to label one of them specifically. The alpha-amino group at the N-terminus will also be labeled by these reagents of it is not blocked. Whether non-specific labeling of surface lysine residue will be useful depends on what you are trying to accomplish.