I've been giving your reply a lot of thought.
As with anything, there are positives and negatives.
Yes, Southern culture is dog culture - - loose dog culture.
When I was biking across southern Alabama a few years ago I went into the the sheriff's office to complain.
When I mentioned the loose dogs, the deputy said, "Jus' shoot 'em." Hmmmm.
BUT!!! These dogs are big chickens. I was solo, you would be in a group of three.
If I stopped and went towards them - they ran with their tails between their legs.
I think that three riders would have few problems - and three riders stopping would scare any dog off.
I don't carry Halt spray - but if all of you have it - I see zero problems.
But you are right - the further north you are - the fewer loose dogs - a direct correlation.
And don't catch a bus - if for no other reason than that bus travel is worse than being chased by dogs.
(They don't call riding Greyhound, "Riding the Dog", for nothing.)
There is no way you will need to make up time - unless there is something major that goes wrong - mechanical or intestinal.
Either US 20 or NEB 2 across Nebraska is like super fabulous.
And nearly all the little towns have places to camp or tiny motels.
Plus a grocery store, a hardware store with a guy who can fix anything, and a home-cookin' cafe.
Nebraska is cycling paradise.
I've biked X-USA six times plus a lot more trips in the West -
What does your tentative route look like right now?
Are you planning to get all the way to the Pacific - or just to Seattle?
Best - J
Pic - Loup River in the Nebraska Sandhills