Meeker isn't on the Flattops and there are no entirely paved roads crossing them. Do you want to go to the Flattops for fishing, boating, hiking, wildlife photography? Is there a reason you can't take the trailer in part way, camp at a Forest Service or State Wildlife Area campground, and drive your pickup into Trapper's Lake?
Oak Creek SWA is near Buford on the paved part of the road and has free camping after mid-July. Bear Lake USFS campground is 15 miles SW of Yampa on a part-paved and part-gravel road and is one of the "Top 100 Campgrounds in Colorado" (from the book by that name). You can then drive into the Flattops from either campground.
If you go the Craig-Meeker-Buford route (CO 13), you may want to stop on the way out to camp at Rifle Falls State Park, one of the state's smallest, but most scenic, and also on the Top 100 list. If you go in via CO 131 (Yampa and Phippsburg) from Steamboat, be sure to try Steamboat Lake State Park first, which is also on the Top 100 list. But both of those SP's are very popular because they have 30/50A electric sites and likely will require reservations to guarantee a camping spot in summer or fall.
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2015 RAM 2500 4x4 crewcab 6.7L CTD; 2016 Rockwood Signature UltraLite 5th wheel