Have you had a flood or fire then?

I had extensive fire and smoke damage a couple of years ago. Instead of going through the hassle I used an independent assessor who dealt with it all. Their fee came out of the insurance claim and through the contractors as consultancy and management. Everything was done well and they did it all. I don't know if this is an option for you. Perhaps ring one and ask about it. It can't hurt to ask if it would cost you or not.

Red and Blue Army!

What matters is that the work is done to your satisfaction. I would get quotes from respectable well known companies if they are on the high side let the insurance company argue the toss.

They can always suggest a contractor themselves if their valuation is nowhere near yours. If push comes to shove I would rather employ a quality firm and pay the difference than employ bogit and sons.

One more point

Insurers employ "Loss Adjusters" to assess damage and quotations for work. Their job is to spend the least money on your claim, and they are complete c-units. It's a steep learning curve dealing with some insurers, so be prepared to stick-to-your-guns on what outcome you want.

I got 3 quotes for a flooded bathroom, and the initial offer from the insurers was 30% lower than the lowest quote for repairs.

I disengage, I turn the page.

yep, the insurers did increase, eventually. But only up to the lowest quote, and that was with caveats about the scope of works.

Don't 'haggle'. If they haven't included anything on their remedial proposal, then remind them ASAP.
Take photos of damage (timed and dated) to record what happened and what they need to fix.

I disengage, I turn the page.

It's just a bit 5hltty the insurers are happy to take your money, but when you have an incident, they really can be a pain.
Loss-Adjusters rank alongside Traffic Wardens and politicians as my most despised vocations.
Unless you have a 'new-for-old' stipulation in your policy, they might also take 'wear' into consideration.

Let us know how it goes, and spill the beans on your insurers

I disengage, I turn the page.