Quotation/Invoice

Meet your client, have a discussion, see how far you can go to accommodated their point of view. They may have to comply to their internal office procedures. Life is give & take.

Maybe I don't understand, but it sounds like they want a second quotation.
If so, the it's up them to get it.
Not your job to run round getting othet companies to quote for work already done....in fact it's not anyones job to run around getting quotes for work already done, it's just a beurocratic waste of time and money.
Tell them you are certainly willing to do it, but will have to charge for the time involved for both you and the third party.
Alternatively just slap them in the face with a wet fish.

Maybe he's just trying to cover his backside because he should have got two quotes in the first place, in which case there should be a 'drink' in it for you.
Del

(The above is subject to terms on conditions based on the fact that I'm a cat and have no place on a construction site, except maybe on top of that nice sunny pile of engineering bricks)

This will require two quotations, but just one acceptance.

1. Tell the customer how much this second quote will cost them.

2. If they agree to your cost quotation for the 2nd quotation, then go ahead and give it to them for the agreed upon price.

Oh, that's what the cat said, too.

I would be very careful here. If you go out to another contractor for a price on this completed work, and if that contractor knows that the work is done and that this quotation is only to justify your price, he may lowball his quotation, knowing that he won't be accepted but that it might put him on the bidders list next time.

If I was the other contractor and I knew that the work had already been performed, I would want to be paid for the cost of producing the comparison quotation. And yes, I might be on the low side to make me look better than you.

"...before we carry out this additional work we have submitted our quotation and client agreed..." This price quotation submittal and subsequent agreement by the client: was your proposal written? Was a written acceptance of the price and terms returned? If these answers are 'Yes', then all is finished. Your client can ask for a new automobile or airplane if he wants to, you do not have to agree.

If you proceeded without a legally enforceable and actionable agreement... well, you have learned a lesson. Your client might be able to make you do all sorts of things you don't want to do. He may, in fact, make you do several unsavory things and then elect to not pay you.

was the addition work done by a third party? i wounld'nt think in would matter, legally speaking. you where the choosen bidder. both of you have a copy of the contract.