Is it allowed in Islam for a man to marry his wife's first cousin, which he knows will cause conflict among them?
Basically there's no shari'a based objection to this marriage.
As what is haram is said in verse (4:23):
Prohibited to you [for marriage] are ... and that you take [in marriage] two sisters simultaneously, ...
The sunnah added the simultaneous marriage of a woman and her aunt (maternal or paternal).
Ibn 'abd-Al-Barr in his Istidhkar added a helpful ruling:
أَنَّ كُلَّ امْرَأَتَيْنِ لَوْ كَانَتْ إِحْدَاهُمَا رَجُلًا، لَمْ يَحِلَّ لَهُ نِكَاحُ الْأُخْرَى، لَمْ يَحِلَّ لَهُ الْجَمْعُ بَيْنَهُمَا
Each two women whom if one replaces one of them (in mind) by a male, a man couldn't marry the other, are not allowed for a simultaneous marriage
So if there's no other objection, for example he already has four wives or he can't afford a second marriage or being just to (all) his wives. By shari'a he can marry her or at least ask for her hand.
Whether he should or not is off topic and can't be discussed here!
Please note that conflicts can arise even in a monogamic wedlock. And if the two wives live in different places and have their "own" home those conflicts could somehow be minimized.
However both the man and the possible future wife should think twice before doing taking this step.