Why go all the way to Whittier? If it's because you have to avoid the Canada border crossing I'd understand, but there's another option.
If you take the ferry to either Haines or Skagway, you can debark there and then drive through Canada up towards Tok and Fairbanks. It'll mean another 300 miles or so on the road but some of the more scenic parts of the AlCan roadtrip are the sections up through the pass from Haines (or Skagway) and in the Yukon between Haines Junction and Tok.
The other bonus is that while Whittier bookings are pretty limited (once a week maybe in summer, none at all in the winter), we get a ferry into Haines almost every day on the summer schedule. If you're already too late to book your preferred travel dates from Bellingham (summer dates fill up very fast) you might look into the Prince Rupert option for your departure port. That also means a trip through Canada customs, since it's north of Vancouver.
You're gonna need a passport eventually so you might as well get one now, and June should be far enough away to use the normal (relatively inexpensive) service. If you're on a PCS move they'll pack and move your firearms with no problems, and you can move ammo (which they won't ship) through Canada in any reasonable quantity with no problems or special paperwork. I'm guessing from the moniker that the health certificates for the dogs aren't going to be a problem

, and you'll need them either way.
SO...unless you or the hubby has a past DUI or felony conviction (which considerably complicates entering Canada) you don't really have to choose between taking the AlCan
or taking the AMHS ferry. Choose the right ports and you can drive some of the best parts while still enjoying the ferry trip up through the very scenic inside passage.
As a bonus, you get to spend more of the trip with the dogs in your lap instead of worrying about them down on the car deck. Honestly, our dog does fine on the ferry but she doesn't really enjoy it either. Even if she does hate travelling, she'd still really rather be together with the rest of her "pack" .