If the OP doesn't wish to say , then that is his prerogative. If I understand the original question correctly - then he was simply asking if he was being unreasonable in thinking that he shouldn't have to pay the extra cost involved.
The simple answer is that if he was given a quote in writing, then , no he shouldn't. If it was a verbal agreement - sight unseen, then he will have to pay for the extra two hours labour. I do not think that this is unreasonable.
A workman is worthy of his hire, and if a job takes a certain amount of time, and a certain amount of materials, then that is what the job actually costs to do. When you employ someone, it shouldn't be that either party wishes gain advantage at the expense of the other. It should not be a game of winners and losers.
It could well be that the reason for the extra cost lies in in the fact that the OP's pluming system isn't configured correctly - something the plumber would only discover during the job.
If there had been a proper contract and it had gone to court, then the court might well have found in favour of the OP . Court judgments might well be legally correct , but they are not always fair. It is up to both parties to act in a reasonable manner before it ever reaches this stage.
I have myself often incurred extra costs when working for customers, because of things that were unforeseen. And, sometimes it is expedient to meet the customer halfway, and share the cost between you.