So when joining a startup (or anything else) that uses Dropbox for Teams, you make a new account just for that project? Perhaps startups (and other entities) that require dropbox membership might make that suggestion?
So (until Dropbox changes something) there does not seem to be a good solution to your question.
It sounds like this was a summer internship, having to lug a separate computer around because Dropbox doesn't support separate accounts seems like a huge work around. The whole point of Dropbox is that the files are wherever you need them.
On the downside, the size of the shared directories would probably push the size limit of your account, so it's not a perfect solution.