I agree - I would definitely not recommend it. I wouldn't even consider that as an option.
I mean, you could purchase a Nexus Pass which allows you to bypass the long lineups, but even then, a Bellingham to UBC commute is not an enviable one at all. Bleh! Rush hour from the border to UBC would be brutal too. It could add hours to your already long commute.
Not only that, but UBC has rigorously implemented a strategy to prevent people from commuting by car. They got rid of most of their parking lots and charge those who use the few remaining lots up to $12 a day. That's not much, sure, but it adds up when you're commuting.
If you're going to look into studying at UBC (which I highly recommend - UBC alumni here

) I would look into living on campus or renting a home/apartment in Vancouver. When you study at UBC, you get a transit pass so you wouldn't even need a car. Transit to/from UBC is fantastic and well-served.
And as mentioned already, chances are UBC's tuition is even cheaper than any tuition you'll pay in the USA, even if you apply as an international student.