To be fair, the triple weave isn't needed for every case and outright is only recommended for 2.48s or faster. Personally I've been preferring the single weave method as something much more reliable with the timing down at slower/base GCDs. It's all about ensuring enough delay after the FBT activation so that the first GCD registers by the Demi.
Bahamut had this problem too, but they fixed it with him. I've had a very old bug report that's been accepted on Phoenix's behaviour from the end of October 2019. It's 8 months later. If it's still an issue come 5.3 I'm going to be fairly disappointed.
Honestly pet gambits need a big overhaul in general. The Summon Order system is a bug riddled failure just as every iteration of pets has had its storied history of broken bugs and exploits.