Hayley,
Just to set the scene I spent about 15 years managing large Contracts for work up to £1bn so I am familiar with Contract Law. So to answer your questions and hopefully help you find a way forward here are some thoughts.
1 : What was your agreed price that you signed up to, and the scope of work that this price reflected?
2 : Prior to the extra work being identified, what (if anything) was agreed in respect of additional or emerging works ?
3 : Was there any agreement as to how additional work / materials, etc would be charged ?
4 : Was anything formally communicated to you about the price changes for the additional work ?
5 : Was the quotation issued with the phrase E&OE or does the "Errors & Omissions Excepted" or similar appear ?
6 : What level of site survey was undertaken ?
7 : In your opinion could a competent contractor have been aware of the potential risk or likelihood of additional work being necessary - agreed anything that might suggest the work would be more involved/complicated ?
8 : Read their Contract Ts & Cs and see what it says about additional works. There may be something that requires prior written authority before incurring cost which you could argue makes the extra costs non-recoverable but this is a thin straw to grasp and may not succeed
9 : What is the Contractual basis of the Contract (IEE or Standard Form, or ICE, etc. Should say in the Contract).
Once you have the answers to these questions then you can embark on the next stage which is to ask them to explain their price.
You should write to either the project manageress, or if not happy at that level, the Operations Manager, state that you are unhappy at quantum of the final cost and ask them for a detailed breakdown of the work they have undertaken, clearly setting out the original materials and labour costs you had authorised and setting out separately the additional labour and materials costs with a justification of how these figures have been arrived at.
As it appears you may well have instructed them to proceed with changes verbally, they will probably argue that it was undertaken on the basis of time and materials or emerging cost. Unless you can prove that either (1) the work was not necessary, or (2) they have undertaken work without your authority then you really do not have a good position in all of this.
You could if you so wish tell them that you had overheard an operative making the £6000 statement but that is hearsay and would not necessarily be useful in any action. They may wish to speak with you and it may well be that they can demonstrate that they have acted fairly.
I fully appreciate all the pressures put upon you but were you to make a claim in Law then this will be looked at dispassionately and will purely proceed on the basis of Contractual Law.
If having obtained a detailed breakdown of the work you feel that the works have been overpriced then you could contact another reputable Company and ask them to give you a second opinion, or alternatively you could speak to your insurer and use the legal protection cover to arrange for an assessor to give you an opinion.
Finally there is a legal obligation on a Contractor to minimise the costs of additional work, in other words they have to act in a professional manner and cannot charge you excessively. There should also be some sort of trade association that can provide details of an ombudsman or an Arbitrator who could be contacted to give a binding opinion.
You should go this route in the first instance before resorting to a legal remedy.
I offer only an opinion above, and how you proceed is a decision for you but you should not proceed solely on the basis of my post here. I would strongly advise you to speak with a Solicitor with experience in Commercial Law at the earliest time if after you have asked them to justify their costs you still remain unhappy. You should give him the information I have suggested you collect above.