Since you said changing ring/pinion and adding trans cooler, I assume you have a 4WD 1500 with 3.08 axle and not the HD trailing equipment. Therefore your tow rating is in the low 5000 pounds range. 3.42 goes to mid 5000 pounds range and with trans cooler/HD trailing you're near 8000 pounds. Cost for those changes is in the $1000 (2WD) to $3500 (4WD) range. If you decide to tow a high walled RV with your Suburban, I highly recommend the trans cooler.
That Wildwood is borderline to much for my 2011 3/4 ton. The 1000 pound receiver limit will be an issue. At 8000 pounds dry, it may hit the 9400 pound tow rating I have when loaded up with full fresh water tank and our typical camping gear. Therefore, no way I would recommend that for your half ton, even with the changes. No one tows an empty RV more than once (from purchasing dealer to home).
Here's what I recommend for you. Find something you like in the 5k dry range and get the trans cooler. Skip the gear change. The 6-spd trans does pretty well, and even with 3.08 axle is better than older 4-spd trans with 4.10 axle. If you can't find something you like in the 5000 pound dry range, consider upgrading the Suburban first. Keep in mind that most half ton SUVs are limited due to payload well below tow ratings. Some with 9000-ish pound tow ratings have 900 pound receiver limits and 1200 pounds of payload. While that may work well for a single guy towing a boat, it does not work week for a larger family towing a travel trailer.