Is it permissible for me to prevent my fiancée from being lenient in her dealings with her sister's fiancé, given my jealousy over her, her rejection of this, and the continuous problems that result from it?

The engagement is merely a promise of marriage. The fiancé is a stranger to his fiancée, so it is not permissible for him to be alone with her, shake her hand, or engage in lengthy conversations with her without need. This laxity contradicts Islamic law and is a gateway to corruption. So, if you are advising your fiancée against being lax in her interactions with her sister's fiancé, this is good. You must also adhere to Allah's boundaries in your dealings with her.