Not exactly.
There is definitely a tradeoff in general between stealth & ordnance load. To be at all stealthy 100% of ordnance and fuel must be carried internally within the stealthy outer shape (“outer mold line” in the argot) of the basic vehicle. Which means either you build a bigger heavier vehicle with lots of hollow space inside for weapon bays or you build a smaller lighter vehicle that carries very few cubic feet of ordnance.
Not directly related to the above, the F-22 was designed as a pure air to air machine. The weapon bays are sized for air to air missiles, not bulkier bombs or air to surface missiles. By keeping them small, they can maximize performance = speed, Gs, altitude, etc.
F-35 came a long a decade+ later and was designed from the git-go for air to air and air to ground. They attempted to square the circle of stealth-induced limited ordnance capacity through a combo of developing smaller weaker bombs, claiming that for limited wars like in Afghanistan smaller bombs were actually better, and trying through precision guidance (AKA “smart bombs”) to ensure a bomb with 1/3rd the explosive would do just as much damage because it would land closer to the desired target.
But even so the weapons bays on F-35 are much larger than F-22 to even hold the baby bombs. And since the airplane is smaller too, that means there’s a lot less of something else in there. Fuel. One of the consequences of the larger weapons bays is less performance by all the other metrics compared to the older F-22. But compared to the non-stealthy predecessor F-15/16/18s the -35 has a pitiful ordnance load and a pitiful unrefueled range due to lacking external ordnance and external fuel tanks.
OTOH, in a modern high tech battle against a near-peer adversary, the F-15/16/18 have a very hard time surviving to arrive over the target with their greater range and payload. Being non-stealthy they were probably detected and shot down 20 minutes ago. While your F-35s have sailed serenely in undetected and / or untargeted to do their dirty work. With their delicate little pop-gun bombs that we hope they can deliver with great precision. While the F-22s help to keep any visually guided enemy fighters off the F-35s. Or at least that’s the theory. Will it work? We may have to find out in the next 5-15 years.
Why no export of F-22s?
The fact the main enemy, the Soviet Union, evaporated a decade after the F-22 project was laid out, but also a decade before the first one flew did not help with the justification one little bit. The development cycle was insanely long. A bit like the B-70 that became the XB-70, and the B-1A that became the B1-B, this was a bleeding edge wonder-weapon that was first deployed after its planned-for enemy had died of old age. The difference was that DoD/USAF was able to re-aim the B-70 & B-1A programs pretty early on to a more useful form and purpose. F-22 came out as it was originally designed / intended. But now with 20 years of unplanned development costs baked in to be amortized over a small fleet.
Once they started to be built DoD realized they didn’t dare export it in case one fell into enemy hands. It was too good, too precious, too techno-magical to risk giving to allies and semi-allies. And too expensive for almost any ally to be able to afford to buy or want to buy.
Not long after that they realized it was also too techno-magical (read “expensive”) for even USAF to afford to buy or operate. So they stopped producing them after fielding a tiny fleet, permanently foreclosing any hope of cost-amortizing export, then loudly started waiving their arms towards the F-35 to distract the anti-DoD faction in Congress from pillorying them over this expensive and largely useless 30-years running boondoggle.
Meanwhile, the opposite tactic was being tried with F-35. Like the F-16 before it, the idea was to sell lots to anybody and everybody who didn’t absolutely hate us this week and thereby get the unit cost as low as possible through the miracle of mass production. Which is in fact working favorably now. Shame about the 15 years of delays and cost overruns getting the v1 product in the air. The first “fully” combat capable version of the F-35 hopes to be fielded in late 2024. Everything out there today is an “interim” configuration with major gaps in the originally promised baseline capability.
Returning to the narrow question of “F”, in a logical labeling scheme the F-22 would be labelled “F-22” and the F-35 would be labelled “F/A-35”. But ref my earlier posts, these labels nowadays are a lot more about inter- and intra-service politics than they are about logically designating a category of combat capability.