Re: fussy baby, tired momma!
Welcome to the forum! We have at least one active mama here who had a baby with a cleft lip and also (IIRC) palate, and I hope she'll chime in soon with suggestions for you.
It's really hard to say when should you throw in the towel and try to comfort the baby in a way other than nursing. It's so individual. I think your "three strikes" rule makes sense, and I'd continue to go that route for now. Just don't worry about nursing "too much", because you can never do that and when a baby has an anatomical issue like a cleft and a tongue tie, the more you can nurse, then better!
Pumping can increase both supply and letdown speed. If fast letdowns are an issue for you, it might be best to avoid the pump for now- though if you're going back to work really soon, some level of pumping may be unavoidable. It's odd that pumping just 1x per day could have had such a big effect- how much milk did you get, when you pumped?
Even in the US, finding a doctor or dentist willing to release a tongue tie can be difficult. Which is odd, when you think about how routinely other medical procedures are performed on babies- you can get your baby's ears pierced or get a male baby circumcised, and no-one blinks an eye- but getting a frenulum released? A lot of docs just won't do it- they'd rather wait and see if the frenulum stretches out on its own, even though as far as I know there are few downsides to taking a proactive approach. I think that the standard for whether or not a tongue tie should be released is whether or not it impacts nursing. It sounds like your baby is getting plenty of milk and gaining weight normally, and you're not in pain- all good indicators that things are as they should be. But a tight frenulum and consequent lack of tongue mobility can make nursing more frustrating for your baby, in terms of preventing him from being able to deal with a rapid letdown.
Coolest thing my big girl said recently: "How can you tell the world is moving when you are standing on it?"
Coolest thing my little girl sang recently: "I love dat one-two pupples!"