Well, I'm not sure the claim "raise M by factor k and P will also rise by factor k" is valid unless one also stipulates that V and Y are held fixed. Indeed, interest rate policy relies on the assumption that movements in key interest rates will shift Y more than they do P, with some of that action arising from a shift in V.
A two-good pure-exchange model (e.g., Edgeworth Box) is a pretty simple, micro-level model that would support the idea that when endowments increase uniformly, so do prices. Take one good as a numeraire, derive the prices in terms of the initial allocations and then increase the initial allocations by some arbitrary factor k. Relative prices won't change, suggesting that individual prices have risen by the same factor (be it equal to, less than, or greater than k). And to homo economicus, relative prices are what matters.
This relies on two aspects of the quantity theory: 1) the economy must be at equilibrium for MV=PY to hold, so that price ratios are already equal to ratios of marginal utility; and 2) increases to M by a factor k are manifested through an instantaneous increase to all agents' budget constraints by the factor k. As far as I know, this second point is only implied by the quantity theory's lack of prescription.