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My 'Ratz' is a challenging print for some machines and slicing apps... but hopefully some of the following helps you get the result you want as it's a very cool print when they come out well.
The mouth is the hardest part the print as it protrudes out on an extreme angle with zero support.... but you can't use automated support as that would lock it up internally. This is the challenge but beating it is the fun part.
They print great on my MakerBot's and the profiles for PLA limit PLA 'without support' at 68 degrees from vertical... so 22 degrees from being back to horizontal. That's a pretty acute angle so I didn't use the limit in this but it is 60 degrees / 30 degrees off horizontal. By rights any reasonable machine / app with PLA should deal with this okay but not all do.
Anyway, here's some ideas:
Make sure you are printing it solid enough. Lot's of people print my designs with less shells and infill than I have recommended. This can make them weak post printing but also not build as well during printing as the structures are too lightweight... they end up breaking down and being a mess. A fraction more filament can help. The crazy thing is skimping on filament can end up wasting more because your prints are too weak / lightweight / not well formed so you end up printing them again with more shells and infill... so you just used much more overall when you could have got it right first time by adding that small amount more.
I appreciate the 'ideal environment' you refer to but that may in fact be working in your favour in this instance. When printing at extreme angles you want your molten filament to solidify real fast... I assume your machine has a filament fan? It's there for a reason... rapid cooling. If your environment is too warm, and the air the fan is blowing is warmish, the cooling will be reduced. If that happens the PLA being printed in this area will stay soft too long and be pushed around as the nozzle does its thing. Try printing cold (i.e. room temperature in the mid 20 degree Celsius range and on blue painters tape).
Assuming your machine has a filament fan you may see an improvement by increasing the fan settings to increase cooling. This is going to make the extruded filament solidify quicker as soon as it's been extruded.
Machine vibration can also affect this... if that's the case it's not going to be easy to resolve as it's inherent within the machine.
The slicing app could also be the culprit. If the nozzle ends up working on that area to long and with a bit too much force it will rip things apart.... the thing to try here is things like rotating the model 45 degrees or 90 degrees to change the tool path. Believe it or not this can help....
Last ditch solution, not that I've done this myself but I am aware of other people doing this to deal with extreme angles.... tip your machine up on the side of the jaw (so the teeth are higher) as this will reduce the angle the jaw is protruding at from vertical. Your machine may be having problems with an angle of 60 degrees but at 50 degrees it could be fine.. so tip your machine up 10 degrees when you run the print. The filament will be less inclined to drop down. This is a bit more extreme but has been reported to help....
Hopefully something in the above helps the Ratz population increase further!. ,