I would suggest getting in contact with a local buddhist temple and see what they would suggest. You won't be able to just show up at a monastery in Tibet and stay there unless plans are made for your arrival. There is a good chance that going to a local temple will get you in contact with someone that will be able to help you on your journey.

The source listed gives you the location of a temple in PA

Spoon is right. You have to get an endorsement from a local Buddhist temple, however, having lived in Laos for almost a year, I have noticed that Buddhist temples are quite open to people who wanted to learn the way. Although I am not too sure if this would guarantee your accommodation, I suggest you take Spoon's suggestion and check out the site suggested above, and this Wikipedia page- List of Tibetan Monastery here:

http://en.wikipedia.org/wiki/List_of_Tibetan_monasteries

I found that the Drepung Monastery is the home monastery of the Dalai Lama. I found out that they are actually open for people who wanted to study Tibetan Buddhist Studies. According to the article, "the training program required twenty years of intense academic study, with rigorous testing and close supervision throughout, as well as six hours of debate each day." I guess your 2 months will not be enough! :) You can find out about them here:

http://www.drepung.org/DLM_Tibet.cfm

And they do have Cultural Exchange Programs that you can find here:

http://www.drepung.org/Programs.cfm

I think your budget of $3,000 (excluding airfare) should be enough for two months of simple living in a monastery.