TLDR: CO2 rise -> Sea Level rise -> coastal city mostly flooded -> Humans engineered marine life that sucks CO2 out of the air super fast and stores it in shells which sink to the sea floor -> Sea floor builds up around city (made of shells e.t.c.) -> CO2 falls -> Sea level falls.
May need to up timeline by a century or two, but with sufficient acceleration of climate swings through human intervention or artistic license could be squeezed into 300 years. (No need to wait till 2117 years before the events start happening as they are happening now!)
Even if you don't believe in man-made global warming, you could for the purposes of the story or invent another cause for it...
Full Timeline:
* Global warming, ice caps melt at ever accelerating rate, Humans, in the effort to prevent global warming, genetically engineer a phytoplankton that grows much faster than normal and sheds lots of calcium containing shells (see Coccolithophores and chalk formation). (Causing Coccolithophore blooms has already been suggested as a way to prevent CO2 rise, minus the genetic engineering bit).
* Sea level rises massively anyway, swamping a coastal city (perhaps over 50m rise if all ice on earth melted). (Climate always lags behind CO2 anyway so dropping CO2 now may still not prevent further short term warming, so very plausible).
* A small river that used to flow through the city is now bigger due to a shift in local weather patterns (very plausible given world climate shift) and water erosion of previously arid lands containing lots of loose soil leads to the river dumping tonnes of silt into the sea over the city.
* The water around the city will be super nutrient rich from all the silt, this would feed the genetically engineered algae's growth massively, shells are dropped around the city with the silt, speeding up its burrial.
* The action of the algae worldwide does lead to CO2 depletion, eventually the temperature catches up with the CO2 levels (even lower than before the human caused spike) enough for icecaps to begin forming again.
* In the far future, this cooling could lead to a snowball earth (if the algae removed enough CO2) but for your setting there could be similar weather conditions to the ones we have now (perhaps even slightly cooler). 279