I've been keeping my eyes open for an air rifle that has the barrel solidly connected
to the receiver so a scope will stay accurate.

Is there such a thing?  (under $1000)

Look for a side or underlever cocking air gun.  Some links for your pleasure www.air-rifle.net www.network54.com/Forum/79537  www.airgunsusa.com  www.airguns.net/aafta/aafta.html  

I have several more if needed.  Love my airguns.

It may seem like it would matter, but really this isn't an issue on anything but the very cheapest models, the <$200 stuff. Barrel-cocking airguns that are made well aren't noticeably less accurate than other types of spring-piston airguns.

Spring-piston guns do tend to vibrate a lot, and firearms-people tend to have a problem getting used to that. If you want something that won't vibrate this way, get a pre-charged pneumatic (charged from a scuba tank).
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I don't know if this would work for you, but I think I'm going to buy one in the future: www.airforceairguns.com. Check the condor.

RWS 48 is a side cocker and a hell of a gun. I love mine.