。
She insisted that he ought to be invited to the party.
Literal
She [topic-は] him [object-を] party [to-に] invite-should [quotative-と] insisted.
A nice stack of grammar in one clause: 招待する → 招待すべき (~べき 'should') → 招待すべきだ (with copula) → 招待すべきだと主張した ('insisted that should invite'). 主張する ('to insist, claim, assert') is a strong verb of stating one's position — used in debates, arguments, and formal discussions. The と here is the quotative marking the content of her claim. Note: with する-verbs, ~べき often appears as ~すべき rather than ~するべき (both are valid).