Just how much is the family weight? With kids still growing, obviously you will want some extra payload available as they grow.
For the bikes, depending on models, I would guess 200lbs for 5 to be safe. Is there anything else in the truck bed? Have you added (or plan to add) any accessories to the truck: spray in bed liner, step bars, truck cap/topper, etc??? Accessories count toward the total payload also.
What axle ratio does your truck have??? You stated you have the 5.0?
From the sounds of it, pending the family weight and other answers, the 28 BHBE may be a possible option..... Part of it will depend on how and what you pack in the pass through storage. Some have reported having a tw of anywhere from maybe 900lbs up to 1200lbs for that model. Do you usually tow with a full fresh water tank?
Check my CAT Scale link which should have a locate link in it and go weigh your truck with the family and five bikes (and anything else you may have in the truck bed) for a more accurate amount of payload that is left for the wdh and the tt tw.
Get a good wdh with integrated sway control. The Reese Dual Cam, Reese SC, and the Equal-I-Zer 4 way system are all very good models. If you choose the Dual Cam, make sure the dealer knows how to install it properly!!! I would probably go with the 1200lb bars as well if you decide on the 28 BHBE. The Reese hitch head (both models) does allow a little more fine tuning of the wd bar angle vs the Equal-I-Zer model, but you can't go wrong regardless of which model you buy!!!
Good luck and keep us posted!!!