The correct word to fill the sentence, with a verb in Infinitive Form is: "No te puedo invitar a la fiesta."
To select the correct word in each case, the context of the sentence and the meaning of the word to be used must be followed, in such a way that once the word is incorporated into said sentence, its general context does not change but on the contrary, it looks better described or complemented.
In the case of the given sentence, which is:
It can be seen that an infinitive verb is required, that is, a verb ending in -ar, -er, or -ir, which corresponds to the context given by the sentence, in which a person apologizes on the phone because they did not can perform an action.
Since the action for which he apologizes is for a party, the only verb in the infinitive that corresponds to that context and cannot be misunderstood is the verb: invitar.
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