My question is, if I can’t find a fourplex for a decent price, would it be viable, or even permitted to construct a building under those terms?

Make an offer on a 3 bedroom 1 bath house in a desirable popular neighborhood that has a finished (non-permitted) basement w/ a full bathroom, sink, separate entrance outside & washer/dryer.

I would be more in favor of grandfathering those already in operation or a limited number of permits given even on a rotating basis.

@Eric Guzman is this being done under permit?

If you bring in the city inspector be prepared to have the home red tagged if you didn’t pull all the proper permits.  

If the answer is no, you do not have the ability to pull your own permits.

What is the difference between "Declared Value" and 'Calculated Value" on the issued building permits in Dekalb county? 

Several (unofficial) sources told me I wouldn't need a permit for a remodel we were doing on our very rural primary residence.

I permit small dogs with breed restrictions, proof of vaccinations, a photo and insurance, with a pet fee. 

Also you sound like your the new guy at city compliance and permitting trying to make things as difficult and expensive as possible for the landlord.