Commercial Insurance Fire Claim: Are you generally required to pay discrepancies in price between two vendors?

I'm the pastor of a church in North Carolina, and this is all real new to me.

We had a fire last Monday, and it was contained in the crawlspace. However, there's a ton of smoke damage inside the main building. An electrical issue (main cable running from outdoor panel to indoor subpanel) most likely caused the fire. The vapor barrier/insulation was charred and melted throughout the entire crawlspace, the wiring was damaged, and the main floor was filled with smoke. The Fire Dept had to break a hole in the outer brick and cut a hole in the middle of the interior floor for access to the crawlspace in a few areas to prevent fire from spreading, but the physical damage was contained in the crawlspace. We're lucky the damage was only as bad as it was… had someone not been onsite earlier than usual Monday morning, it probably would have been significantly worse.

The insurance company sent a third-party adjuster onsite Monday afternoon and gave us the A-OK to start with getting contractors / etc., for the repairs. So, we did. On the first day, we had the following contractors onsite: Electrical, HVAC, Media (media install was damaged), and Restoration/Damage Remediation. We had quotes from all of them by the end of the week, and the electricians had done enough work to get some outlets turned on so the remediation company could start ASAP

All contractors sent the quotes to the third-party adjuster, and the damage remediation company 'seemed high.' So, the insurance company sent a preferred contractor onsite Monday morning. He was $11,000 cheaper ($42k vs 53k) but the quotes are not Apples to Apples. In fact, some of the things itemized on the first contractor's bid are literally just left off of the second contractor's quote, and we cannot get the adjuster to understand that. They're saying if we use 'our people', we'll be on the hook for the extra $11k. But, they also told us to get somebody in there ASAP, so 'our people' have already installed Air Scrubbers / done some initial cleanup.

What's the best way to fight this / counter this? We're not trying to do anything crazy here, we literally just want our church fixed properly. Our electricians, HVAC, and other locals in our area recommended the first contractor. The second contractor can't start as soon, they are located 1.5 hours away, and we have no record of the quality of work they do.

TL:DR: Insurance Company says we have to pony up $11k to continue using the contractor they initially told us to go ahead and call in. How do we push back?