We have lived near the mountains in the western Denver suburbs for 50 years. But we have camped at both Chatfied SP and Golden Gate Canyon SP so the grandkids could experience sleeping in the trailer when they were young. If the state parks are booked, try the campground at Bear Creek Lake Park, a City of Lakewood park with 3 lakes, boating, swim beach, playground, etc. It is a couple of miles south of I-70 and near the intersection of US 285 and C-470, so convenient to everywhere in the metro area.
We seldom go near Estes Park anymore because we can't stand traffic jams when going to the mountains. When we visit RMNP, we camp at the Forest Service Stillwater cg on Lake Granby. It has electric hookups, water, and a dump, is less than 10 miles from the west entrance to RMNP via US 34, and near my favorite scenic area of Middle Park--Monarch Lake.
And snow may not only be available in RMNP for the OP to play in, it could even be available in Denver. Our sons played soccer games in snow in a Memorial Day tournament 3 miles from our house!
For more info on camping, routes, and not-to-miss sights in Colorado, see my blog
HERE.
PS--There are NO Corps of Engineers campgrounds in Colorado! Lots of Corps lakes, but their campgrounds are all state parks. And the state parks just raised their campground fees by a significant percentage on January 1. Plus you must now have reservations--no walk-ins during the primary camping season.