This is interesting. It could work if, like you say, you sanitize the heck out of it. But, tell you the truth, I wouldn't use it. Do you have a glass jar of jam, jelly, spread, juice, etc? Any food you have in a glass jar could be emptied and you can use the glass. Actually, you could use a beer bottle (or two), just cover them with foil and you should be good to go!
If were you, I would go to the grocery, go to the natural foods section (might have to go to a Whole Foods like place) and buy a gallon of apple juice in the glass jugs. Pour the juice into the milk jug. Clean the glass juice jug. Use the juice jug of your starter (and many more to follow) and drink the juice (it is good juice).
If there is no time for that, I would use a 22oz beer bottle. I would worry about getting all the milk smell/taste out of the plastic, but other than that, it should work fine.
I ran into a similar problem. My small stopper kept sliding out of a 22 oz bottle because I used star san and it was still wet. I went back to the first bottle I had used with my first starter which was originally a 1/2 liter coffee creamer container. I had bleached the f* out of it and then rinsed for no less than five minutes until the smell of creamer and bleach was completely gone and then I applied star san and let it dry. The first batch has no infection and my third batch (second with this container) is also contaminant free so I would say that as long as you are rigorous in cleaning it you might get a bug but I wouldn't count on it.
+1 on the glass jar, and as an aside I'm not sure how useful the starter will be after just one day; if it's a smack pack, you generally let it swell for a day before pitching into a starter in the first place. After that, you want the starter to work for at least a couple of days. It least that's my understanding from reading and that's how I've done the couple of starters I've made.