Claiming off the third party direct (or their insurers) is a claim in Tort - that is you have to show negligence. This presumably has been accepted so now the question is the extent of damage that resulted from the negligent act and cost to put you back in the same position as you were. Inevitably they will question your claim and seek an alternative opinion - again they have done this. Nothing much you can do until you get that second opinion. If it agrees with your engineer OK, but otherwise you will need to consider what compromise is acceptable.
Doubt your insurer will pay more given the same reports, but the advantage is potentially less hassle. The difference is your claim is under contract which limits what they might pay - for example if they have a "new for old" clause you may well find you get less.
As always, it is the detail that is important and you can only decide the best course of action when you have all the facts.
