Hmm... Yeah, I sort of went through a similar thing when I applied for permits.
I had to include a site plan that accurately showed the location of the house, driveway, sidewalks, utilities, storage shed, gazebo and the proposed structure.
I then had to calulate
Impervious Cover. I think I was allowed 65%.
I ended up with 42%, which is good because I plan to put in a pool at some point. So in my area, we too are also limited on the amount structure you can put on a lot.
My property is sufficiently large enough that I am allowed to have a "Guest House".
Now we get into the nuances:
A guest house is of course considered a separate dwelling unit and as such plan review fees and inspection permits are considerably higher than a "detached addition" which is what I was applying for. I siad to the reviewer, this isn't a guest house, there's no shower or bath, just a sink and toilet.
He said, "But you have a kitchen too." I said, "Yes, but its just going to be a sink and refrigerator".
His response was, "Technically, that and the toilet is all you need to be considered a guest house."
I said, "Well, what, are people going to bathe in the sink!?"
Anyway, after further discussion, common sense prevailed, and he agreed to permit it as a detached addition.
I guess the morale of the tale here is that there are always instances in city codes where you may not be able to permit one thing, but by making ever so slight changes to a plan, you may be able to permit it after all. Being a civil engineer, I make my living reading and understanding city codes, and how best to develop properties under those codes.
I'd get a copy of your city's development code, and search through it, and find out what the actual criteria for development is. It could be as simple as something like your just 50 sq.ft. over the allowed limit.