Fantasy book with a magic female lead who is being hunted, and is traveling with a guard and a guide

Question

The woman has special powers, and is fleeing from her homeland, from which she is banned. She is traveling with companions who are helping her cross a "haunted" mountain. She finds a strong red-haired ax-wielding guide who has a wolf/dog that hunts. One of her "guards" is from the King's assassin group, but is protecting her with his life.

They cross the mountain and are attacked by a horrible (invisible) animal that is known to kill anyone crossing that part of the mountain; she saves the guard and they continue their trek.

They are supposed to cross during the day to avoid this entity, but have to fend this animal off. She is trying to make an alliance with an enemy before she is captured.

Answer 1

Found it! The Shattered Court by M.J. Scott:

Entangled in a court ruled by tradition and intrigue, a young witch must come to terms with newfound power and desire—and a choice between loyalty and survival.…

The royal witches of Anglion have bowed to tradition for centuries. If a woman of royal blood manifests powers, she is immediately bound by rites of marriage. She will serve her lord by practicing the tamer magics of the earth—ensuring good harvests and predicting the weather. Any magic more dangerous is forbidden.

Lady Sophia Kendall, thirty-second in line to the throne, is only days away from finding out if she will be blessed—or perhaps cursed—with magic. When a vicious attack by Anglion’s ancient enemies leaves the kingdom in chaos, Sophia is forced to flee the court. Her protector by happenstance is Lieutenant Cameron Mackenzie, a member of the royal guard, raised all his life to be fiercely loyal to the Crown.

Then Sophia’s powers manifest stronger than she ever imagined they would, and Cameron and she are inextricably linked in the process. As a witch unbound by marriage rites, Sophia is not only a threat to the established order of her country, but is also a weapon for those who seek to destroy it. Faced with old secrets and new truths, she must decide if she will fight for her country or succumb to the delicious temptation of power.…

Answer 2

This is probably not the answer, since it's only a partial match, but what about To the Haunted Mountains by Ru Emerson?

This is the first tale of the Nedao people, told from the perspective of Nisana, and AEldra of the cat kin. The story is left to her, as one of the few witnesses of the trials of the brave young queen Ylia, the Lady of Nedao. As she trains Ylia in the arts of her powers, Nisana is the only mind to have full access to all of Ylia feelings and emotions as she is thrown into exile from the city of the King and subjected to dark magics and a dangerous journey through unfriendly lands.

It has an exiled female with powers who is headed through the eponymous haunted mountains, with an animal companion, and is hunted. That said, I don't see any mention of the two male companions.