I certainly wouldn't miss Red Lodge, MT (nice place to stay too) as the NE entrance to YNP (+ Tower Falls) and Tetons are among my favorite spots. I to would try to stay in the park, as it would be very memorable. If it is just a night, I would do a lodge in YNP and a Lodge in TNP. I'm not sure the cabins would be practical for a night, but they are pretty fun and 'period' / possibly romantic if they are available. I just saw a special PBS on Jackson Lodge, and the pancake breakfast on Jackson Lake via horse drawn wagon ride looked like a 'Must DO' for kids.
I would avoid Jackson for Lodging, but that's just my style. Jackson lodging is better for mom and dad getaway, or skiing. But consider something really rare for kids (less plush for Mom and Dad)
Be advised on travel, It can take a long time to transverse the park in Aug... 25 mph...(but you have been there, so you should remember.) Animals + tourists + kids + cameras + narrow roads and NOW Radar

(glad that came AFTER my motorcycle days)
Are you driving in from an Airport, or any particular direction? There are some interesting routes from different approaches. (If you are helicoptering in, that would be great too!)
From the East / NE I like 14a from Sheridan area through Burgess Jct. & the Bighorns
If you are coming from an area near Laramie, be sure to cross the Snowy Range and take a dip in the free "Hobo Pool" at Saratoga.
Our kids always enjoyed Thermopolis water slides (Once they overcame the smell). I would usually sleep under a shade tree while they swam, cuz I preferred to drive at night while they peacefully slept. We did lots of city parks w/ pools to let them stretch while traveling. (Like the "Lusk 'Plunge'

, that's been awhile... ) Something that was real fun for them was squirt guns for August trips through WY (since we never had a vehicle with AC). The VW camper van was perfect, but slowwwww. Very noisy, so driver could 'ignore' the kids, with 'on-board' sink to refill the squirt guns ! and fridge for treats + upper and lower bunks (The kids definately remember those trips, 25 + yrs later... they were both in diapers at the time...)

Cloth diapers....
