Winger Fitzharris, 19, spent last season on loan at Cappielow from Celtic, but was recently released by the Parkhead side.
He has, the Tele understands, snubbed interest from SPL sides and offers of trials from clubs in England to sign a one-year deal with Ton.
And Frenchman Bachirou has ended speculation he may not return to Cappielow by putting pen to paper on a new one-year contract extension.
Yesterday also saw former Hamilton defender Andy Graham, 27, complete the formalities of his deal after returning from holiday.
Moore told the Tele: "I can confirm we've signed Sean Fitzharris. He has agreed a one-year deal and is a player who I know exactly what I will get from.
"He's a local lad and lives in Inverkip, and I think that's been an important part in us being able to entice him to come back here.
"He's a winger who excites fans and gets them off their seat with his dribbling and ability to run at people.
"I played in the same position, so I know exactly what I want from my wide men. I like to play with a 4-4-2 or 4-3-3 and for them to be able to beat men and get crosses over, and Sean has that ability.
"The only thing with Sean was that he's suffered a wee bit with injuries. I'm hoping that he's over some of the problems from last season and we see more of him on the park next season as he can be a big player for us."
Moore is also delighted to secure the signature of fans' favourite Bachirou, who had previously cast doubt over his Cappielow future after admitting suffering from homesickness during his spell in Scotland.
He said: "Fred has also signed a new one-year deal and returns on 20 June for pre-season training. He's a very good player who the fans really took to because of his tenacious playing style, especially towards the end of the season when he was getting a game every week.
"We're starting to build something here for next season, so we're delighted he'll be with us for that."
And completing the trio of signings is Ton fan Graham, who Moore has captured at the second time of trying, also on a one-year-deal.
He said: "I had Andy with me at Stirling and he's the best pro I've ever worked with.
"I tried to sign him last year but he decided to join Hamilton to play in the SPL.
"As soon as I knew he was available again, I had no hesitation in going straight back out and signing him.
"He's from Dunoon and is a big Morton supporter who has spent many a day watching the club from Cowshed, so he'll be delighted to be coming here.