There is a principal that "You are allowed to change for the sake of peace."
The source for this comes when the messengers tell Avraham that he is going to have a child in a year. Sarah laughs, saying, " God changes Sarah's statement from "After I have become worn out, will I have smooth flesh? And also, my master is old." (Gen. 18:13) to And the Lord said to Abraham, "Why did Sarah laugh, saying, 'Is it really true that I will give birth, although I am old?' (Gen. 18:14).
Rashi comments on this change: although I am old: Scripture altered [her statement] for the sake of peace, for she had said,“and my master is old.” - [from B.M. 87a].
Another source states how Aharon would privately approach each person in a fight and tell them that the other person wanted the fight to end. Using these white lies, Aharon made peace, Avoth dRebbi Nosson (ch. 12).
There are limits to how far this concept can be applied. If one is testifying in court, of course, then you need to tell the truth. I do think that you would be allowed to "change the truth for the sake of peace" to avoid a grudge.
There are two other issues that you should consider:
The truth may (will?) come out. If you are in a situation where you did not want to use a specific business, and instead you went with a different provider, there is a good chance that someone will find out that you went with the different provider and tell person A. It may be best to go to the person directly with your concerns.
I saw that you asked for sources. These questions are great to consider in the abstract, and to compile all sorts of different opinions. But when an actual situation comes up, having the chance to talk to a real person who knows you and can help you apply the sources is invaluable. Very often the sources seem to contradict each other, and a balancing act is needed to weigh what is correct in light of Halacha and pleasing in the eyes of humanity.