I suspect the Specialized bike you linked to would work OK for a loaded tour. Might not be ideal though. Gearing is 34x30 low. Not super low for a loaded bike. You could probably change the cassette and go lower but extra hassle. Since your trip is a loaded tour with camping and cooking and you are carrying everything, it makes sense to get a loaded touring bike. Surly LHT, Trek 520, REI Novara Randonneee are all candidates. $1200-1500 in the US. All are identical. All will work fine. Racks, fenders, bags, super low gearing is all possible with these. Make sure you get super low gearing when you buy the bike. You already have a Specialized racing bike. So might as well get a strong, do everything loaded touring bike for its partner. Use it for commuting and groceries when you get back home. And riding across the USA here.
A loaded touring bike is probably what you want for a loaded tour. If after you start you decide to go ultra light and carry minimal gear, that is good. The bike will still work. You won't notice a problem riding it. Once you get some bags on a bike, they all ride the same. 10 pounds, 20 pounds, or 30 pounds. They all ride sort of bad. So having a light bike or a heavy bike won't matter. When I say bad I don't mean awful, horrible, terrible. Just bad as in not quick and agile and light. Its got pounds of bags hanging on it. Its no longer a quick lively bike. Its a beast of burden. You're not going to sprint up the hill carrying luggage on the bike. So whether its the Surly LHT, or Cross Check, or that lighter Specialized bike, it makes no difference.