Forming ester from amide using protonated alcohol as catalyst
There are several points that I'd throw out:

As a technicality, you probably can't buy protonated alcohols; you'd need to use the alcohol plus an acid like HCl in a non-aqueous solvent.

Now, once the amine drops off, it will mop up your catalytic acid, so you had probably better use at least stoichiometric acid, and possibly quite a bit more.

Amides are more stable than esters; you know that already. So, the reaction thermodynamics are not in your favour to begin with. Thus, in order to get a decent yield, you probably also need to use a big excess of alcohol (Le Chatelier's principle: more alcohol → more ester).

You'd probably also need to heat it quite a bit to get the reaction going. (The same is true for hydrolysis with aqueous acid.)


All in all, probably you could get it to work, but it might not necessarily be fun, and if your reactants had any other sensitive functional groups on it they would pretty much be destroyed by heating in strong acid.
In general, this is a tough reaction, and I suspect that if you could find a general way to do this cleanly then it would be quite a substantial achievement. Recently, Hie et al. have developed a Ni-catalysed procedure (Nature 2015, 524 (7563), 79–83), but it only works on aromatic  amides, so isn't applicable to your example substrate. (Essentially, both the group attached to the carbonyl carbon, as well as one of the groups on nitrogen, have to be aromatic. Despite that obvious limitation in scope, it's still in Nature, which maybe suggests something about the difficulty of the transformation.)