If transferring your role to a new city would result in a significant pay bump, then I think it’s either an expansion of your role (promotion), or else it’s just a locality adjustment as the cost of living is more expensive in the new city. Since you think this move would knock some years off of your path to FI, then I’m assuming it’s an expansion.
In which case, it sounds like you’re capable of a more advanced role that pays more, but you’re only considering this one opportunity that your current company is offering. And unfortunately this one opportunity would require a move that would lose social connections for each member in your family.
Instead of considering this move, can you take this offer as motivation to find a promotion within your current city? Are there any other companies in your current city that are hiring for a similar expanded role?
Or, if your company wants to promote you to this new role, maybe they value your work enough to figure out a way to have you do it without actually moving? Possibly commuting one day a week to the new city but the rest of the time telecommuting?
I’m coming at this from the view that you don’t want to move. But only you can determine that. If your kids are young (in elementary school) then they shouldn’t have much trouble making friends in a new place, so I wouldn’t weight them much in the equation. However, if they are in junior high, then it’s harder, and moving to a new high school is really miserable. For your adult social circle, think about where you originally met your current friends. If most of your social circle is from a group that you won’t have access to in the new place (like if your friends were all from college), then you likely will have a hard time building up a similar social circle. But if all your friends are coworkers, and you’ll be working in a large office in the new city, then you’ll probably make a similar cohort of friends at the new office.
As for how your company reacts to you turning it down, well that depends on your company and how much they value you. If they’re a big corporation and there’s lots of competition for promotions, then yeah, it’ll probably hurt your career progression if you end up staying at the company but not taking the new job. But on the other hand, I have had companies react the opposite way when I’ve turned something down, especially when I’ve turned it down after reaching FI. Basically, if you’re valued, then they actually start to value you more when you turn things down, because they realize that you have options and they start not wanting you to decide those other options are more attractive than their position.
