You can selectively protect the e-amino group of lysine (e.g. with Boc or another carbamate protecting group) in the presence of an excess of Cu2+ (the alpha amino and carboxyl group form a bidentate ligand to the Cu2+). Please consult a couple of papers for exact reaction conditions.
In page 4 you can find the synthesis of our compound, I would still use the protection of the amine provided in the previous paper given it's simpler and this particular paper's objective is avoid metal catalysts, something that shoulnd't be a concern for you, and pH control is way easier to fail in comparison to Cu2+ chelation. Basically, I'd do the Boc protection according to the previous one and the acetylation according to this one.