If your hitch is only 10k and your trailer is 12k then this conversation is moot. There's a lot of things you can go over on and fudge a bit but hitch ratings are not one of them. Oficially,, since you have a slushbox, your truck is rated for that 5er ( why they rate a truck with a shushbox for a 12k 5er and handshaker only a 11k 5er , I'll never know , Kind of seems backwards, probably has to do with the dual mass flywheels
but you're playing with fire running 2k over on hitch capacity on a 5er and if you drop that sucker in the middle of a mountain, those mounties are going to want to have a serious talk with you
Now get your hitch situation straightened out and she'll pull it fine. Flush out the tranny and change it out to some kind of synthetic tranny fluid and as mentioned, have the biggest honkin cooler you can get, one of those remote coolers with its own fan so its not heating up your coolant more is a great thing.
If you are going to be towing heavy on a regular basis , a Rodney Red radiator is a worthwhile investment