The BVRAAMs (in the F-22's case, it's the compact variant of the AMRAAM, the AIM-120C) take up a lot more space than the AIM-9X close-combat missiles, so the Raptors will be sacrificing their short range combat capability by a lot.
On the other hand, if the Fulcrum pilots decide to screw their shitty BVRAAMs and stick with their plenty capable short-range missiles, they will most likely kick the shit out of the outnumbered Raptors even after taking losses from the AMRAAMs (which, I should mention, are not Jesus-missiles... they miss, too). Remember, the MiG doesn't always carry all missiles, but if they DO choose to do so, and load up to their wingtips with AA-11s, which are just as if not more maneuverable than those Sidewinders the Raptors are packing, they'll be loaded with almost twice the number of missiles the F-22s have. In short, even after a few Fulcrums have bit the dust, the rest can just FILL THE SKY with missiles, all of which are racing towards our friends at Mach whatever. Even the Raptor's high stealth won't help when each pilot is being overloaded with missile launch profiles every five seconds... they'll be too busy evading to even get off a shot. Sooner or later (okay, sooner RATHER than later), quite a few of those Archers will score a hit and we can kiss bye-bye to one hell of an expensive airplane. Or a few of them. Or... a LOT.
You make those Fulcrums Su-47s (which, with their utterly superior maneuverability and utterly inferior stability could probably outmaneuver quite a few of those AMRAAMs), toss in the Berkut's superior weapons load capability (they have internal bays PLUS external hardpoints, which the F-22 doesn't have) into the mix, and you've got yourself a very embarrassing and very expensive dogfight for the USAF. Even if the Raptors DO win, which they very well could (depends on how good those pilots are), the USAF would have lost a lot more money than the Russians/Soviets/Ultranationalists/Chinese, which makes it a logistical defeat rather than a military one. Remember, missiles aren't free, and those AMRAAMs cost a fortune.
If I were you, I'd stick with loading the F-22s with Sidewinders... and, oh, scrambling the nearest squadron of less costly F-15Cs to nail those retarded Fulcrums (or Berkuts) who are too busy chasing the Americans' legendary Eff-Twenty-Two. F-15s, when you get down to it, are no slouches in dogfights (in fact they kick ass like there's no tomorrow). Oh, and they don't cost as much.
Scratch that. The best option is to be a good Boy Scout and be prepared... with a whole lot of Avengers placed on standby. I mean seriously, how can you miss 150 non-stealth jets coming right at you? So the moment those retarded little Russkies pop up, you give them a hearty welcome with half a million surface-to-air missiles (which they have no way of fighting against, having loaded up with Archers instead of air to surface missiles). They'd be stupid not to retreat... that is, if they still have enough planes left to follow any semblance of a command chain.
In conclusion, you can't just hope to throw F-22s at anything and hope to come out with anything close to a complete victory. The successful repelling of a sizable enemy threat requires mastery of multiple resources available to you, as well as the versatility to break lines when needed to trick the enemy into pushing too far and depleting their resources. Any commander must see all the options at hand - say, do I have jamming units on site? Is a diversionary strike to draw away a part of the enemy force feasible? Should I retreat and draw a more prepared ambush to destroy the enemy force? and so on. Force and technological superiority will not win battles, commander!
Dismissed!
(If you haven't noticed, I didn't actually think all this up on my own. There's a guy who lives across the hall who wants to go to West Point... I asked him because he knows more shit than I could hope to ever think about picking up.)