From page 5-52 of the 2010 Yukon Owners Manual the tow ratings for your year SUV are about:
3.08 axle: 5000 pounds3.42 axle: 5400 pounds
3.42 axle with HD trailering package: 7900 pounds
As far as I know GM never offered the 3.08 axle with the HD trailering package on the 2009+ SUVs.
To confirm your equipment, there is a sticker in the glove box with a list of 3-character RPO codes. You will have one of GU4 or GU6 for 3.08 or 3.42 axle. You will have Z82 for HD trailering package or not have it listed at all
I highly recommend adding a large capacity transmission cooler for towing any high walled RV if you do not have the HD towing package. At 5k-ish tow rating, you'll want to stay closer to 4000 pounds from a manufacturer's dry weight rating. RVers add stuff, lots of it. Bikes, chairs, tents, toys, games, food, cookware, utensils, clothing, etc.
If you have the equipment for 8000-ish pounds, then 7000 pounds seems like the high end limit. However, you may be limited by the 1000 pound receiver limit or your vehicle's payload. If you have 1400 pounds available and the family weighs about 500, then you only have 900 left for trailer tongue weight.
Performance wise even the 3.08 shouldn't be that bad. It is very similar to the older 4-spds and 4.10 axles which few people complained about for 7000 pounds. You'll have 4 usable towing gears.
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