I had a house fire back in 1995 which only had fire damages in 1 bedroom and part of the roof, but the major problem we had was smoke damages that caused the entire house in the inside to be stripped bare. The restoration company was there the next morning, thus they took everything, like clothing, furniture etc to a place that they tried to (remove) the smoke damages.
The key with a restoration company being a flood or fire, wants to (save) as much, thus insurance saves money on not paying for claims that they can save due to their restoration. A good restoration company knows that the insurance will pay them, be it 30 days or more, (unless) insurance disputes your claim, then you, since as the owner had to sign to have the restoration company to try and mitigate your damages.
For the time being don't worry if they pay them or not, since if insurance disputes, then they will have to send you a denial letter of the reasons of (why) they dispute your claim, then at that time the restoration company will be after you to pay the bill.
You are getting ahead of yourself, vs just focus on your claim with the insurance and my guess they will settle with you and if not, then since you have hired an independent adjuster whose job is to make sure your claim is paid and if not, then to fight for you.
Retired liability adjuster but (never) handled homeowner claims