In a formal sense, on paper, yes it would work now, and is probably satisfactory for a first-year organic class. It does have a couple of problems from a laboratory point of view that might be worth considering...
1) MeBr and MeCl are both toxic gases and are difficult to work with. In fact, the major use for methyl bromide is as an industrial fumigant to kill insects. There is a much more convenient, cheap alternative that is almost always used when you need a methyl nucleophile.
2) Chlorine free radical halogenations are not very selective - you will get a mixture of compounds mono-chlorinated at each available site, as well as dichlorinated species, and so on. You might want to consider a somewhat less reactive halogen for that step.