Tree growing into gutter. What's a good strategy?
It's hard to say from that photo, but it looks like that bulge is the stump of a limb that could be trimmed back further. Really that's a question for an arborist, though. If you do so you'll want to pick the right time of year to help prevent fungal infection. 

I agree that you're probably misinterpreting your city's ordinance, though. No sane city government would prevent a homeowner from protecting his dwelling from invasive plants.