Is it really so outré to write "posso ti invitare"? ti is the object of invitare, after all, not of posso.
You should say "posso invitarti ?".
Yes, or "ti posso invitare". But I was marked wrong for "posso ti invitare", which makes more sense to me (and would be the correct order in French).
in spanish we would say te puedo invitar..... the te is equivalent to the ti and it goes before just like in the italian
Like you, I initially found it counterintuitive, but (at least according to the Italian Michel Thomas course I took) it really is wrong. The pronoun can go before the modal verb or at the end of the infinitive, but never between the two.