People towed Hondas for years when the manual said "No". Some other brands too. Obviously involves some warranty risk, but that 2013 is mostly out-of-warranty anyway. Post-warranty, the risk is to your wallet.
The 2013 Civic shares the powertrain with other 2013 Hondas that are designated as towable (CRV and Fit) models, so it probably could be towed successfully. The fact that Honda refuses to approve it ought to give you pause, but if you are a gambler it's probably not a bad bet.
Automatic transmissions suffer increased wear & tear when towed, and some are more vulnerable than others. Manufacturers seek to minimize their warranty exposure so place limits on towability, but marketing likes to expand their sales potential with promises of 4-down towing. Somewhere in the middle is a gray area of moderate risk, too much to promise warranty coverage but maybe not too much to be done (with caution). Holding the towing speed down, limiting the time under tow, and occasional stops to exercise the tranny can all mitigate the potential for damage. Your mileage may vary...