a quick check would be to calculate the headloss in the system from the catch basin all the way to the river. add up the friction loss in the pipe (assuming flowing full), add losses for flow through manholes, bends, junction structures etc. and add entrance and exist losses at the beginning and end of the run. add this number to the water surface elevation and compare to the rim elevation. This could be done in a spreadsheet very easily.
This is done assuming full flow for simplicity. Also, since you have water backing up, full flow is a good first assumption.