Your partner can use an ITIN to purchase a home. However, it is going to be more difficult and require more work. There is some good information here: http://www.usimmigrationsupport.org/immigrants-realestate-itin.html.
Your best bet will be to start with your bank/credit union and/or big bank and discuss this with them personally before filling out tons of applications and hard inquires on your credit reports. The best bank may be the one that is seeing the $3500 a month go in consistently. Sometimes, Credit Unions will issue loans and/or credit on a case by case basis taking everything into account..
You can read more about ITIN Home Loans here:
(Note that things have tightened with the credit crunch so you will need to talk to someone for the most up to date information.)