Unfortunately I don't see any sysvar or ActiveX object which states the block object / -name opened in BEdit. I can't even see if there's anything by which to check if BEdit is active.
So my guess is - nope, not possible.
You could go an redefine the BEdit command so you can obtain the block's name. Then run the undefined ._-BEdit to open it. But that means you'd also need to recreate the dialog and the double-click functionality so it appears to the user as if nothing's changed.