I'm not sure what this place actually costs, what your budget is, or whether this would accommodate everyone, but you might consider the
Robert S. Kerr Mansion in eastern Oklahoma. I went there a few years ago as a part of historical survey and was blown away by the place, both the place itself and the surrounding area. I remember one of the people with us asking what the place cost and recall that it seemed fairly reasonable to me at the time. But, I don't remember the specifics. The mansion itself is built on a hill and surrounded by a lot of nothing but hills and trees and absolute beauty.
It's near a small town called Poteau, in the Ozarks, and is close to an even smaller town called Heavner where the Heavner Ruin Stone is located. If you're not familiar, this is a stone discovered with ruins on it of apparent Viking origin, suggesting that Vikings traveled around what would be come the United States, up the Mississippi, and into the tributaries that eventually run through modern day Oklahoma, possibly as far as north as a little south of Chicago. There's a small museum there as well as the stone itself, which sits protected near where it was found below a cliff. That, in my view, is a must-see. And, if you can't stay at the Kerr place, if you end up anywhere near there, go. The
Spiro Mounds are relatively close as are several lakes, streams, and rivers suitable for fishing.
Poteau also prides itself on the "worlds largest hill" nearby. It's just shy of the official definition of a mountain, i.e. still a hill, but pretty big for a bump in the ground.
Whatever the case, my main suggestion would be to find some place in Southeastern Oklahoma. More natural beauty there, imo, than anywhere else in the state unless you like FLAT.
Regarding the weather, don't even try to plan around it. March *tends* to be mild and windy, but we've had both ice storms right up through April as well as ferocious thunderstorms and highs in the 80's at the same time, sometimes within days of each other. Summer is hot. June and early July *tend* to be more reasonable than late July and August when it can and usually does reach near 100 or more. For the sake of your health, wherever you choose your lodgings, pick something with hotel like qualities, i.e. rooms with a roof and A/C unless you're all adventurers who don't mind throwing dice with the weather and can handle what can be downright atrocious conditions that make the animals retreat to caves and higher ground. The weather might be perfect, but you really don't want to get caught out in a rustic camping ground when a flash flood comes along with no "home base" to which you can retreat. Many a camping trip when I was a kid was cut short by tornadoes. Tents don't hold up well against them. :-)
P.S. If any of you are drinkers, bring it with you. The beer is 3.2 (water) unless you go to a liquor store, where you have to get it warm. And, finding a liquor store can sometimes be a challenge in that part of the world. Plus, many counties in SE OK don't allow restaurants to serve liquor. I only mention this because I got yelled at once by a friend who'd never been to OK and went to a restaurant I suggested on a drive through because he couldn't order a glass of wine with his meal. I forget sometimes most of the rest of the world isn't like this.