It sounds like a seriously cool and worthy trip - and, possibly, a great road trip as well. Unfortunately, I can't offer helpful advice or tips with regards to an exact route between Seattle and Jackson Hole because a) I've never done that exact trajectory myself, and b) it's been some time since I've driven around Washington State, and even longer since I drove up to, through, and around Jackson Hole. So why am I even trying to give advice on a trip that I haven't actually made? Simply because almost all of my road trips through that part of the world - including Washington State and parts of Idaho and Wyoming (and possibly even the northeast corner of Oregon) have been memorable and generally excruciatingly beautiful - so I can unhesitatingly say that I would HIGHLY RECOMMEND renting a car and making an extended road trip. With 16 days, that theoretically leaves you a decent bit of time for some cool side trips 'en route' - but the truth is, on a road trip, one never has enough time. Of course, you have alternatives: you could do the road trip on the way to Jackson Hole and then fly back (doubtless the most expensive version, as one-way car rentals and one-way plane flights tend to be pricier); or you could do a 2-way road trip, there and back from Seattle, but take different routes each way. I think this last would be my choice - as it would allow you to see more; though the practicality and scheduling need some planning.
Among the many, many cool possibilities is the one you mentioned - Mt. Saint Helens in Washington State. (Confession: I've lived in Oregon for several decades and have always wanted to make a trip of several days to Mt St H, and never done so yet. Damn! You are inspiring me.) There are several ways to access it but supposedly one entry point, from the north, allows you to get closer to and access some of the more remarkable hiking trails which truly let you experience the mountain.
Heading south from Mt St. Helen's, you reach the mighty Columbia River, which is long, large and magnificent and divides Washington and Oregon. If you started at the westernmost point - in the charming small town of Astoria (on the Oregon side of the river, and home to nice restaurants and brewpubs and coffee shops), you could spend a day or two or more simply going eastwards along the Columbia. The Columbia Gorge (more on the western side of the state of Oregon) is a national scenic area with wonderful hikes, waterfalls, etc etc etc; the town of Hood River is a mecca for windsurfers because of the high winds which go across that part of the river. I think there may be quite a few other spots along the Columbia worth seeing, too.
Another potential side-trip would be cutting down across the northeastern corner of Oregon, home to the Wallowa mountains which are splendid, rugged, and isolated. (2nd Confession: I've been yearning to make a Wallowas-only trip to that part of the PNW or Pacific Northwest, and still haven't gotten around to it. Damn!).
Probably, your route will take you through or across southern parts of Idaho, near the city of Pocatello. There are beautiful rivers there, and the area is scenic. (And also, rather socially and culturally and religiously conservative, though the past decades have brought a sprinkling of more liberal-minded dwellers, as well as more nice coffee shops.) I think parts of southern Idaho are meccas to paleontologists - certain locations are or have been treasure troves of fossilized dinosaur bones. Not sure if any are 'visitable', though.
Eventually you will cross into Wyoming and reach Jackson Hole. Once, many many decades ago and in a previous lifetime, I went on a memorable road trip which took me up to and through Jackson Hole - I still remember how stunningly beautiful it was - and also how cold it was. My trip was made at the end of Fall/beginning of Winter, which I think is significantly later than your projected September dates; hopefully you won't be running into heavy ice or snow. (But if you were, an AWD or 4WD vehicle could be a lifesaver.)
Coming back to Seattle, one cool side trip is going to the San Juan islands, in the upper Puget Sound. Accessible only by Ferry, from points north of Seattle, they are isolated, rugged, and beautiful. It's also possible they might be touristy and crowded. Parts of the northern Puget Sound are beautiful. Another trip near Seattle is the Olympic Peninsula, on the Northwest point of Washington State. Parts of it are a primeval green dripping rain forest - other parts, including small waterfront towns (like Port Angeles, along the inland Salish Sea that divides the U.S. and Canada) can be either charming, picturesque or touristy, depending. But the Peninsula itself has a world of possibilities - worth considering if you have time.
Short version of all of the above: I truly think you could have an epic road-trip.
Most of my recent trips to parts of that part of the world have been to parts of the Columbia River - or parts of Washington State - not to mention a ;pt of Oregon - but I'm by no means an expert so you may get more up-to-date tips from others. One final thought - there are many great camping spots and campgrounds, both in Oregon and Washington, that are created and maintained by the respective states of OR and WA - so checking out the State Campground websites (and possibly for Idaho as well) would be a good idea. In the lengthy recent pandemic period, going camping (in beautiful isolated spot)s seemed to be one of the few things people felt safe doing -- so nice campgrounds tend to get reserved months and months and months in advance. Moral of story: it's a good idea to explore them now, well in advance of your trip.
And...I didn't even mention Yellowstone. Whew. All things considered, 16 days, which sounded like a lot initially, may not be enough for ALL OF THE ABOVE.
But... it sounds like a wonderful trip.