Chain suck is really sucky. But to many chain suck occurs while shifting, not usually refereed to when there's been an incident/fall down. Chain suck is usually thought as the chain failing to release off one chain ring while shifting. This causes the chain to get pulled up and around the crankset on the "backside". In the extreme the chain contacts the chain stay and can get wedged between the rings and frame. Is this what you are saying?

You should have the chain ring teeth checked for wear and/or bent teeth & wobbled rings. The chain should be checked for a twisted link. If all are in good order then look at the front der adjustment and whether you're soft pedaling during the shifting. This last point is very important. many riders don't have the best shifting technique and power during the shifting. This hard pedaling during shifting only makes any existing condition much worse.

Sure you can ask the LBS to install the chain you already have. Don't be surprised if they have a higher labor rate for customer provider parts installs. Also remember that the performance of that chain will not be their problem. Andy.