Yes, that に and the causative verb are linked. After removing less important modifiers, the sentence is basically this:
私に恋人契約させた男子!
the guy who made me lover-contract!
the guy who made me sign a lover's contract!
So 私(に) is the agent (causee) of the past causative form of the suru-verb 恋人契約する. Also note that this sentence is a 体言止め sentence.
And:
- 私 is modified by a relative clause, 大魔王の娘である ("who is a daughter of the great demon lord")
- 恋人契約させた is adverbially modified by その巧みな話術で ("with his smooth talking skills")
- 男子 is modified by an i-adjective, 抜け目ない ("shrewd")
Therefore, the final translation is:
大魔王の娘である私にその巧みな話術で恋人契約させた抜け目ない男子!
(He is) a shrewd guy who made me, the daughter of the great demon lord, sign a lover's contract with his smooth talking skills!