Call Remco and ask them - they will tell you straight, won't try to sell you things you don't need.
As far as I know, the 2003 Civic can be towed like any other Honda automatic - just follow the standard Honda directions, which are:
** Instructions from Honda on towing 1989 and later Hondas **
Check to be sure the transmission fluid level is full.
Release the parking brake.
Start the engine.
Shift the transmission from Park to Drive.
Shift from Drive to Neutral.
Turn off the engine; leaving the key in the Accessory (I) Position
Make sure the radio and all accessories are turned off.
NOTICE: The steering system can be damaged if the steering wheel is locked. Always leave the key in the Accessory (I) Position to prevent damage to the steering system when towing behind another vehicle.
CAUTION: SEVERE AUTOMATIC TRANSMISSION DAMAGE WILL OCCUR IF THE CAR IS SHIFTED FROM REVERSE TO NEUTRAL AND THEN TOWED WITH THE DRIVE WHEELS ON THE GROUND.
We do recommend starting the engine and shifting the automatic transmission, if so equipped, through the gears approximately every 500 miles and shifting from Drive to Neutral before shutting off the engine.
For all manual transmissions automobiles, release the parking brake, shift the transmission to neutral, and turn the ignition key to the Accessory (I) Position to release the steering wheel lock. Make sure that the radio and all accessories are turned off.
Although experience has shown that the Honda/Acura transmission and powertrain are capable of being "motorhome towed", American Honda does not assume responsibility for any towing vehicle, lighting hookup, or other towing equipment or towing procedures. And responsibility for these items is assumed by the Owner/Operator.
Former owner of 2004 American Tradition and several other RVs
Home is in the Ocala Nat'l Forest near Ocala, FL
Summers in Black Mountain, NC