The engine is up to it. I have a 2007 Ridgeline with the 3.5 J35 engine and I have exceeded the recommended tow capacity with about 6000 lbs of trailer weight once plus many other times pulling close to the max 5000 lbs. There were no issues at all towing the weight. Gearing, suspension and brakes are what limit the Odyssey. Your transmission cooler is a excellent addition for towing and I would second a weight distributing hitch. Since you said you are new to towing you need to remember to keep about 10% of the total trailer weight on the tongue. Insufficient tongue weight will cause the tail to wag the dog. Scary if it happens and you don't expect it.