Well first thing is that you can not hunt squirrels in Texas with an Air Rifle, there was a whole big discussion not to long ago on that note......Second lots of break barrel air rifles users use scopes, I do...
because the front sight moves every time you cock the gun
I guess that says a lot for the engineering of modern day break barrel airguns. I have several and all are glassed. VERY solid and accurate shooters. That same statement would hold true for irons too, not just scoped movable barrel guns. In theory, the barrel would never return to the EXACT same position, so the irons would be inaccurate also. So if the statement were true, you should NEVER be able to hit consistantly with a break barrel. And we know that's not true.
Your heavy power specs are (to me) unrealistic for a young shooter to enjoy and spend the time to master. It takes time.You may want to read up on the ballistics of the .177 vs .22 because ALL airguns use the same powerplant for both calibers. The ACTUAL FPE will be very similar in both. At 25yds or better 177 heavy pellet goes deeper. Just like same powder charged .22 v .38Cocking effort,LOP, and Accuracy, would be my first concerns, over advertised power. A good shooting 12fpe is a 35yard rabbit & squirrel killer! Our EU friends been doing it for years at 50yds.A good spring will cock easier for him(progressive), and unless you spend big bucks on any "NP" you don't know what it's power is for sure. They are mass produced and not consistent. And when they leak, that's it... A spring is cheap and easy to fix.The "Artillery Hold" is the common amoungst airgunners.
For an air rifle with a gas spring with an adjustable trigger. You may want to work backwards in your research. Since everyone swears by a Charliedatuna trigger upgrade for smoothness and adjustability in a trigger. His website shows all the air rifles it can be used on. May find a gas spring air rifle with open sights. Not sure of the quality of the open sights that will come with the air rifle in your price range. LOLJust had a fast look at Pyramid Air website. Looks like all the gas spring air rifles that have open sights either will take a CDT trigger and or the Hatsan gas spring rifles have a Quatro adjustable trigger.