Is it permissible for the suitor to speak on the phone with his fiancée after only the recitation of the Fatiha, knowing that the engagement will last two years and the marriage contract has not yet been concluded?

It would have been better to complete the marriage contract (nikah) so that meetings and communication could take place without legal complications. As for speaking with the fiancée, if it is done respectfully and with assurance that no temptation will arise, then there is no objection.