Possible? No...See my last sentence above.
The Library name (and Linkage ID) are properties of the Library, which is NOT compiled into your SWF. It would defeat the whole purpose of Flash and the Small Web Format (SWF) concept if this was to occur.
However, you can use JSFL to dynamically identify which clips are instances of a Library item, rename them, etc., but only within the Flash authoring environment. JSFL can interrogate the Library properties...for example, the following code will display a dialog box containing the Linkage Identifier value for the selected Library symbol:
The following code will apply an instance name to a particular symbol (in this case, a symbol residing on frame 1, layer 1, of the main timeline:
You can see that objects on any timeline are stored in an array so, just as with Actionscript, you can also iterate through your symbols; rename them; increment the instance names; or whatever you want....while editing your FLA. But you won't be able to do the same with an SWF in the Flash Player. In the above code, the "fl" object represents the Flash application, the "documentDOM" object represents the FLA, and the "library" references the Library Panel; none of which exist within an SWF.
fl.getDocumentDOM().getTimeline().layers.frames.elements.name = "newInstanceName_mc";