Lee, who is an assistant coach and keeper trainer at Kaizer Chiefs, was roped in on an interim basis to help Bafana prepare for a 2019 AFCON qualifier against Libya on Saturday at Durban's Moses Mabhida Stadium.
Stuart made the decision after current Bafana goalkeeper coach Andre Arendse told SAFA two days before the team went into camp that he could not attend as his wife was suffering with complications from surgery.
Stuart, who worked briefly with Lee at Turkish club Genclerbirligi and SuperSport United, is aware hiring his son over the likes of Mamelodi Sundowns keeper coach Wendell Robinson and Arthur Bartman of Maritzburg United could attract accusations of nepotism.
However, he insists he made the call with Bafana's best interests at heart.
"I have to say that, I'll be massively, massively surprised if there is any reaction whatsoever," Baxter said about Lee's appointment at a press conference at the Hilton hotel in Durban.
"Because Andre pulls out at the very last minute, Lee is the most qualified coach in this country by a mile. Lee has worked with me in Turkey and at SuperSport United, and [AmaZulu fitness trainer] Joshua Smith in the technical team. He's the current coach of Itu [Khune]. He has worked with 'Ronza' [Ronwen Williams] and he knows probably half the squad.
"We don't have time to go wandering around the country interviewing people, bringing people in. There's no one really else I've worked with.
"For the good of the country, for the good of the squad, I don't ... the only thing I think we were concerned about is if Chiefs, given their situation, would release him now. They have been good enough to do that.
"So if anybody sees anything else other than it is the best and most viable decision I could make, I'll be massively surprised.
"Both Ronza and Itu actually petitioned the football association in getting [him to] camp.
"Of course you know that people are going to have opinions about that and they're going to say, 'well, no, he's got his son in, this and that'.
"But Lee is the best qualified goalkeeper coach in the country by a country mile.
"So I don't expect there to be a lot of opinions about the selection, but I do understand where you are coming from."
Bafana started their first training session at 15h00 at Kwamashu's Princess Magogo Stadium.
SuperSport duo Bradley Grobler and Dean Furman were excused as they are nursing knocks, while Bongani Zungu, Themba Zwane, and Hlompho Kekana have been ruled out of this weekend's encounter with injuries.
Baxter has since called up Matsatsantsa utility player Aubrey Modiba, Orlando Pirates winger Vincent Pule and Sundowns midfield workhorse Tiyani Mabunda as replacements.