"Need to balance historical/cultural sightseeing with active fun kids stuff since we will be traveling with our 8-year old daughter."
We took our daughter to the Mosel at the same age. I question whether Beilstein will meet any of the needs you mention. It is perhaps the smallest town on earth with very little to do - a honeymoon sort of place. I think you'd be better off staying in Cochem, where there are lots of diversions (chairlift ride to lookout, mustard mill, etc.,) shops, bakeries, open markets, and interesting cobblestoned streets and alleyways to explore at any hour.
It would be more convenient for you to stay 3 nights in one place than to split them between Bingen and the Mosel area. If you're worried about being tired, that's very understandable - it's not a good condition to be in when driving in strange places. Instead of driving, I'd suggest taking the train straight to the Mosel. Then pick up your car in Trier when you visit there (a short train ride away) which should also help you avoid airport pick-up fees.
There are several connections to Cochem where you can hop on at the airport, make one change in Koblenz, and your there in 2.5 - 3 hours' time. If you stay awake, the scenery is fantastic and no one has to drive.
The train ride to the Mosel will be quite cheap - 27 Euros for all three on a "Rheinland-Pfalz" ticket daypass (good from Mainz onward) plus a few Euros each for FRA - Mainz (not covered by the R-P ticket):
http://www.vrminfo.de/vrm/english/sites/tickets/rp-ticket.html
On Sat or Sun, you can use the "happy weekend" pass instead:
http://www.bahn.co.uk/db_uk/view/products/weekend.shtml
(Use the regional and local trains - RE, RB, S and TR trains - with these discount tickets. To find these trains, use the following page and tick off "only local transport" under "means of transport".)
http://www.reiseauskunft.bahn.de/bin/query.exe/en