You could try either Paragon Partition Manager (I believe there is a free version you can download) or the imagex utility in WinPE.
However, since the objective of disk imaging is to create a working backup, all of these disk imaging programs are going to throw their toys out of the pram when they encounter disk issues.
Frankly, I would strongly suggest that you mount the defective drive as a slave on another system and use a program like Robocopy to copy the entire content to another volume. You can set Robocopy (free download) to perform multiple retries before it gives up and goes on to the next file.
That way you have at the very least, a substantial backup of any application and data files.
After that, you can attempt a run at CHKDSK, which will normally correct most issues without causing more problems. The time when chkdsk does cause issues, is when the hard disk is already so defective that it is beyond reasonable recovery anyway.