There's a big difference between the ability to do something and the ability to do something safely. I would not feel safe doing what the OP is wanting to do. Even if you drive safer and more slowly to compensate, there are variables that occur every day for which your risk increases.
How many times have you been cut off, had someone pull out in front of you or simply had traffic lock it up in front of you without warning and with no lane to change into? How many times has this occurred to you and an accident only barely been avoided. Add 5-800# and new geometry (both the trailer length and pivot point loads during low traction situations, not to mention the affect on your suspension) and the risk seems too great for anyone to seriously consider doing this over buying an appropriate vehicle or renting a truck for less frequent needs.
That should be easy to tow. No need for trans cooler. Dad used to pull a 1500lb trailer with a 71 hp car with no issues all over the country. No engine or trans issues. Car was scrapped due to rust at age 16.
2007 Corolla Red Pearl 155k miles PP 0w20 & ST 4967
Tow Capacity for the 06 Civic is apparently 1500 Kg [3306 lbs]*
That is the complete, all inclusive, load capacity ... passengers, fuel, cargo, trailer, and whatever is on the trailer.
If it's an Automatic Transmission, leave it in Drive. DO NOT haul that load in an overdrive gear. Ever.
"Good, Cheap tow packages" don't really exist. You want a proper trailer lighting harness installed on the towing vehicle.
Check with your local regulations (and every State/Province/Country you plan to travel through) as to how heavy a trailer must be before it requires trailer brakes. That's the trailer weight itself and whatever is on the trailer.
If you need them, you will need the proper interface installed on the vehicle. Finally there is proper installation of the hitch in the correct Class on the vehicle. Because the Civic is not a body-on-frame construction vehicle, installation might be more involved than otherwise, but it's certainly doable.
Finally, if you want answers, please provide complete relevant information. Vehicle make, model year, and in this case, engine and transmission (manual or automatic) all matter. Plus, there are a lot of motorcycles that are under 500 pounds, but there are a lot that are not. 800 pounds for the bike alone is certainly possible and remember, if you stuff things in saddle bags, they count as cargo weight. So, what motorcycle?
Better call them first, then. They will refuse to rent if their computer doesn't think your vehicle rates. It's been a while, but they de-rated my Dakota by 2,000 lbs. Their computer ratings were ridiculous.