Is it permissible for a Hanafi to adopt the Hanbali maddhab regarding the an oath of obstinacy (Nathr al-Lajaj), by offering an expiation for breaking an oath (kaffarat yamin) instead of fulfilling the vowed act?

Vows of obstinacy and anger are treated like oaths. This is when the one making the vow intends to urge himself or others to do something, or to prevent something, or to confirm or deny a piece of news. If someone is a Hanafi and makes a vow of obstinacy, they may adopt the Hanbali madhhab in three cases:

First: If they seek a fatwa from someone who gives them a fatwa according to the Hanbali madhhab, because a layperson is not bound by a specific madhhab; rather, their madhhab is that of the one who issues them a fatwa, and their duty is to ask the people of knowledge.

Second: If fulfilling the vow would cause extreme hardship or unusual difficulty, they may adopt the other madhhab to alleviate this hardship and difficulty, as ease and the removal of difficulty are considered legitimate Islamic principles.

Third: If it is to follow the stronger evidence, provided that the person is a student of knowledge capable of understanding the proofs and their ranks, and they switch from one madhhab to another for a religious reason, such as when it becomes clear to them that one opinion is more preponderant than another. In such a case, they will be rewarded for it; in fact, it is obligatory for them to follow what appears to them to be closest to Allah and His Messenger.

What is forbidden is switching between madhhabs merely to follow dispensations without evidence, or imitating another scholar, or a valid Shar'i excuse, as that would be following one's whims.

In summary, it is permissible for you to act according to the madhhab of Ahmad regarding your vows, or to revert to your own madhhab. Your matter will be properly regulated by seeking a fatwa from a scholar you trust in their knowledge, or from an established scholarly institution that issues fatwas. This is the obligation of the layperson and what their affairs depend upon.