Maybe easier to at CH3-S- to 3-Bromine-alanine. Br-CH2Ch(NH2)-COOH
This seems like a very bad idea...
1. I don't think it would be particularly easy to brominate alanine, so not sure where the SM would come from
2. It would probably form an aziridine if the amine wasn't protected (and possibly even if it was, i've had NHBoc trap onto adjacent leaving groups)
3. The bromide would probably leave in an E1cb if you exposed it to base
I think methyl iodide+cysteine sounds much easier. You might even get away with not protecting the nitrogen, sulfur is a much better nucleophile. I found many examples, e.g. JOC 1985, 1264, or if you dont have access, Beilstein J. Org Chem 2012, 1105 (open-access).