You can't just use MeNH- because you need some kind of counterion for the minus charge. Then you run into the same problem you have above where you have to lose that counterion somehow. Imagine that counterion is H+ and you have the original case.
Nonetheless, as others said, there will always be some base around to deprotonate the SN2 adduct. In problems like these, you have to assume that you have all the reagents you need to get to the product. You'd have a very tough time isolating the direct SN2 adduct in this case. That said, if you were to react a tertiary amine (e.g. Et3N) with something like methyl iodide, you would make a quaternary ammonium salt (Et3MeN+I-) which could be isolated. In that case, however, you can't just pull a proton off of the N though.