If I understand correctly, let's simplify slightly: I have a pot that I used to cook regular milk (and nothing else) year-round; can I use that same pot for Pesach?
Okay let me simplify that question one more step: "on Purim, I took a brand-new pot. I poured in a gallon of water, tossed in one grain of wheat, and cooked it all for a few hours until the wheat had totally dissolved. then I washed it out really well. The pot sat, cold, clean, and empty, for a month. Now can I use that pot on Pesach? Or um, let's say it was one kernel of corn instead of wheat, and I'm Ashkenazic. Now can I use the pot?"
That's what this boils down to (pun intended). That year-round milk may contain trace amounts of chametz. (More likely vitamins that are kitniyos, but if it's on a line that's also producing malted milk, let's say trace amounts of chametz.) Before Pesach, such chametz is nullified 1:60 by volume. On Pesach, it's not.
You could buy such milk before Pesach, at which point it's nullified, and even keep it in the fridge and drink it on Pesach. You should not, however, go to the store on Pesach and buy milk that may contain, let's say, 0.1% chametz, because on Pesach, it's not batel.
When it's all said and done, standard practice is that if a pot was used for any amount of chametz -- or even kitniyos -- before Pesach, we kasher it before using it on Pesach. Which is probably your best solution here; clean it out well, wait 24 hours, then run the whole thing with boiling water.