I wonder if ethers are more or less acidic at the alpha position than alkanes ($\mathrm{p}K_\mathrm{a} \approx 50$). I'm not sure if the stabilising inductive effect from oxygen or the alpha effect (the repulsion between lone pairs on the carbanion and the adjacent oxygen) will dominate.
Based on $\ce{H2O}$ ($\mathrm{p}K_\mathrm{a} \approx 16$) vs $\ce{H2O2}$ ($\mathrm{p}K_\mathrm{a} \approx 11.5$), I would expect the inductive effect to be stronger, but I would like to check.