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.
I agree with mmathis' comment that the ideal solution is to present the issue to the city to deal with the tree. The tree will only continue to be more intrusive.
That being said, if you have to deal with the tree as is, the best solution would be to shorten the gutter by either cutting it and installing a new end cap or replacing that section of gutter with a shorter one. You will want to install a diverter (also called a shield) on the roof. This is simply an "L" shaped piece of metal. One leg of the "L" will slide under the shingles and the other will stick up. You will want to install this at an angle to kick the water from the edge of the roof over to the shortened gutter. Be sure not to allow debris to build up, but given the proximity of that tree, you are probably already cleaning you gutters very frequently. Something like this photo.
Sooner or later, this tree is going to do more damage to your house than bend your gutters. Its roots are under your foundations, and as they grow they will eventually crack them and cause subsidence. Talk to your city about your options, and get an expert to deal with the tree, sooner rather than later, as the more damage it does the more expensive it will be to repair.
A strong wind could push the tree closer to the gutter, and an earthquake could shake either the house or the tree so they hit each other hard, so severe damage to the gutter is possible. I suggest pointing that out to the tree's owner.