Is subscribing to a marriage website under a pseudonym to track a husband who is also subscribed to the same websites considered unlawful (haram), and what is the expiation for a false oath made during registration?

If "bypassing the oath" means you succeeded in overcoming it without performing it, this is not permissible if it was a condition for entering the site, due to the Prophet's (peace and blessings be upon him) saying: "Muslims are bound by their conditions." You must repent, which is expiation for this sin.

However, if it means you did what was asked of you (writing the oath or clicking on it), then there is a difference of opinion regarding whether a written oath is valid or not, and clicking is less weighty than writing.

Your entry into this site, if it is for searching for spouses, is meaningless and will not help you in knowing your husband's plans. It could also be a pretext for bad things, and it involves prying into your husband's private affairs, which is not permissible according to Islamic law.