Double checking towing capacity on '08

Hey everyone, new to the forum, and hoping someone with a little more expertise can help clarify whether or not my truck will be able to tow what I need it to this summer.

The specs:

2008 Ram 1500 SLT Regular Cab 4x2
5.7L V8 Hemi
5-Spd Automatic 545RFE Transmission
3.55 Rear Axle Ratio

The trailer I'll be pulling has a base weight of 4200 lbs., and gross towing weight shouldn't exceed 7000 lbs. This is where I get a little hung up. According to Ram Body Builder I should be fine towing up to 7850 lbs. with "heavy duty cooling", but without that I can only safely pull 5550 lbs.

With that in mind, my spec sheet also lists an Auxiliary Transmission Oil Cooler as part of the truck's optional equipment. Is this what's considered "heavy duty cooling"? Are there any additional steps I should take to ensure my tranny doesn't fail on me this summer? Note that I'll be hauling 1500+ miles.

Thanks a lot for your help!

My 2003 Reg cab 4x2 2500 lists the same optional cooler. My rear axle is rated for 6000 lbs. Front: 4000 lbs. My class 4 receiver hitch is rated for 1000 lb tongue weight, and gross trailer weight of 10000 lbs.
Your build sheet will tell you the class of your hitch.
Highly recommended to download your build sheet. I am certain the limits for your 1500 will be lower than my 2500z

What class hitch do you have? It will determine the max weight trailer you can tow.