The first place Alexius would have turned to would be the Wardens.
Even if the leadership remained silent after Alexius promised funds, recruits and an ally in the Magisterium, all it would take would be a Warden with fewer scruples for him to learn the secret.
Afterwards, it's a matter of convincing them to place Felix through it and even, possibly, allow him to be discharged of his duties. Again, funds, recruits, etc.
Here's the problem: the exact process of becoming a Grey Warden is not actually common knowledge among anyone outside of the Wardens. There are very few people who know that the Joining can - in theory - save someone who is already Blight infected. I'm sure that people are aware of Warden immunity to the Blight, but they don't know how the Wardens get that immunity. Just because the player has knowledge stemming from DA:O doesn't mean that everyone in Thedas knows what you know.
The only reason Carver/Bethany can survive the Blight infection and join the Wardens is IF you actually bring Anders along, because he is a Warden who knows about the Joining process. Even if Alexius had thought about trying to consult with the Wardens early on (before he signed on with Corypheus), he had no actual reason to believe that becoming a Warden could have saved his son. Again, nobody outside the Wardens knows that it's the Joining process that gives them their immunity and that it can save a Blight-infected person; as a result, I think he probably dismissed the idea out of hand (mistakenly believing that it was too late for the Wardens to help).
As for Alexius having Fiona around, by the time he had her and the Circle mages in his pocket, he was working for Corypheus and believed that he was going to save his son through serving Corypheus. Even if he had been aware that Fiona was a Warden, Alexius had already settled on a course of action. He had no reason to consult her or talk about his family problems at that point - she wasn't a friend or true ally, she was just a tool.
It's unfortunate, but love can make you do stupid things (like believing an ancient Tevinter magister is going to somehow save your son).