i have lagered in a hdpe plastic bucket with no problems. both the better bottle (i think are pet) and the hdpe are oxygen permeable but you are not likely to notice. i would be more concerned about the oxygen in the head space. when i have done it i have moved the entire bucket near the end of fermentation in to the fridge with the air lock on, that way at least i am reasonable that the head space is co2. yes it was on the yeast still.
also thickness matters in terms of diffusion rate. those spray cylinders made for spraying decks and yards are also thicker hdpe usually.
some breweries now are using hdpe kegs (plastic kegs of america?) with little difficulty