However, what is the situation like if one of the parties is a non-practicing Catholic (baptized but not much of a believer)?
Well, first, this is a very bad idea.
As to what the law says on the matter:
Can. 1071 §1. Except in a case of necessity, a person is not to assist without the permission of the local ordinary at:
4/ a marriage of a person who has notoriously rejected the Catholic faith;
§2. The local ordinary is not to grant permission to assist at the marriage of a person who has notoriously rejected the Catholic faith unless the norms mentioned in ⇒ can. 1125 have been observed with necessary adaptation.
Regarding what constitutes “notorious rejecction” it is someone who has consciously rejected the Catholic faith and that rejection is publicly known. It would be up to the pastor to examine the situation to determine if the person had “notoriously rejected” the faith. If so, then paragraph 2 would apply and the norms of mixed marriage would apply.
(Note: notorious rejection is not the same as formal defection. Formal defection meant actually writing a letter to the bishop. Notorious rejection requires no such act, it is based on the 1917 code and its definition of “notorious”).
Also, if the person completely rejects the Church’s teachings on marriage, then they cannot be married. From Familiaris Consortio:
However, when in spite of all efforts, engaged couples show that they reject explicitly and formally what the Church intends to do when the marriage of baptized persons is celebrated, the pastor of souls cannot admit them to the celebration of marriage. In spite of his reluctance to do so, he has the duty to take note of the situation and to make it clear to those concerned that, in these circumstances, it is not the Church that is placing an obstacle in the way of the celebration that they are asking for, but themselves.
So a case like this must be carefully investigated by the pastor, pastoral council given, and then a decision made by the pastor as to whether or not the bishop needs to be involved.