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Do a "compression test" or even better do a "Leak Down test" on your engine. This will point you towards the problem cylinder and or head. The leak down tells more about the health of the engine than a simple compression test.
Like lotsofspareparts above I would probably do it myself and save a wallet full of money in the process. The question is "How mechanical are you?" If you have some experience with a small block Chevy it would be a plus. Get the Factory Shop Manuals for your specific year Corvette and follow the manual as it will show you how to do it. Frequently it might only be one cylinder head gasket that failed but you will probably have to machine both heads to keep the numbers the same on both sides of the engine. This makes the job much more involved as you need to remove, machine and re-install both heads.
As for the cost of the work, Here in the Washington DC area there is no way for $1000. If you live someplace rural that is a different story as the prices might be better. Here in our area they are charging $175 an hour for a decent mechanic. Find a good machine shop and they can help you a lot with their specialized knowledge.
I pulled both heads off of my 1968 C3's 427 and they were out, fixed and back on the car in a few days. Those BB cast iron heads are HEAVY, yours are aluminum so you won't need a second person to help too often. When my Cadillac STS blew a head gasket I junked the car instead of dealing with the Northstar engines, aluminum heads and aluminum blocks make it a very hard job.
$2,000 to $2,400 where I live in West Virginia for a good shop to do the work. That's for both heads , the price you were quoted must be for only doing 1 head... I'm no master mechanic, and I replaced head gaskets on my 89 in 4 days, working about 6 to 7 hours a day, but I worked slowly and carefully...Doing it myself cost me around $400.00 , that includes Machine work on the heads, Felpro gaskets, and ARP head bolts....WW
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Are the LT1 blocks made of aluminum or are they still cast iron? If the block was aluminum I would suspect the price will go even higher as the steel bolts get corroded into the aluminum and are not removable without pulling the threads out with them. I built performance air cooled engines for many years and remember always having a set of heli-coils ready as using an aluminum case we were pulling threads out all the time. I ended up thru bolting them so they would stay together better and longer.
On the L98 it is much easier with the cast iron block and aluminum heads like on my 1988 C4. The one thing I do is use anti-seize whenever you have steel bolts or even spark plugs in an aluminum heads.
If you trust the shop doing the work , and they're charging that low of a price, I would go for it... I really don't see how they can do it for that price , I checked with 3 different shops and the best I could do was double that price... If you do a search on this site you will see most saying around $2,000 for the job...WW