Okay, some progress. No base, no reaction. So, even though NaOAc may not generate much anion, but presumably it was the sodium salt that reacted, no? I recognize that one cannot just add a base, except maybe NaH. There will also be additional agents, e.g., acetic acid.
Since methanol gave the best result, what happens if base is added slowly to the reaction mixture with BzlBr present?
From the description, you aren't finding the result of reaction of the alkylated product are you? If so, that would explain why a stronger base would leave to lower yields, not because the reaction failed, but because the product is more labile in the reaction conditions.