Is the marriage mentioned in the question considered a "badal" marriage and unlawful, taking into account the questioner's fears about its effect on her marriage?

If the marriage in question is not a shighar marriage, which is forbidden, then it has no relation to it. However, since the described characteristics of the suitor indicate that he lacks both religious commitment and good morals, he is not suitable to marry your cousin. This is based on the Prophet's (peace be upon him) saying: "If someone whose religious commitment and character you approve comes to you seeking marriage, then marry him. If you do not do so, there will be tribulation on earth and widespread corruption." You must advise your family, and if you do so, you will have fulfilled your duty. If the marriage proceeds, pray to Allah that He makes it good, and frequently ask Allah for well-being so that this marriage does not negatively affect your own marriage.