I think that legally, you probably have every right to have the tree chopped down. I wouldn't mess with that big root without talking to the city first. You should be able to give someone in your city a call and have a conversation about this. If not, you can probably talk to a realestate lawer and have them research it for you.
The tree is very pretty. I think that cutting that large root would jepoardize the overall health of the tree. I would take a recipricating saw and pruning blad to the root ends that are touching your foundation though. At the very least, that should stop them from growing further into the foundation. I've cut a number of roots in my yard (much further from the trees, and the trees are much taller). The saw takes care of them with ease. You can jam the saw right into the dirt.