Malolactic questions

I am doing my first Malolactic fermentation. It is on 6 gallons of Chardonnay, pail juice. I did the primary fermentation in a plastic (food grade) bucket using Red Star, Montrachet yeast, stirring once a day. The primary fermentation is just ending so I added the Wyeast, malolactic cultures and stirred the bucket of juice. My questions are 1.Do I leave it in the plastic bucket during this malolactic fermentation till it is done, if so, do I stir it regularly or leave it alone? I am concerned about exposure to air. 2.Would I be better off putting it into a 5 gallon glass carboy with an airlock now and just leaving it alone for a month or two and then racking it? My temp in the basement is a constant 68 degrees. I will add oak chips after the first racking.