Is it permissible to speak with a non-mahram woman through an intermediary (her friend) out of necessity, if the aim is to avoid temptation from direct conversation?

It is not permissible to continue corresponding with this girl or exchanging news with her, even if through an intermediary or a mahram. This is to block the means that lead to temptation. If the correspondence is for the purpose of engagement, then the marriage proposal should be made to her guardian, followed by the marriage contract.