Sheesh. If TM can't handle LW, TM is more unreliable than I've heard.

Best bet would be to remove the LW folder from /Applications, and then create a new admin user account on your Mac, and see if TM will properly handle a brand new user account.

If so, then you know the problem is in your existing user account.
To remove all the LW setting from your existing account, open the home folder's Library, and search for "lime"

That will find you the home/Library/Preferences/LimeWire Folder and the home/Library/Preferences/com.limegroup.gnutella.plist

If you really want to find ALL LW related files and folders, you'll also want to backup/delete the Saved, Incomplete and LimeWire Store Purchased folders.

Lastly, search the hard drive's Library (not just the home folder's Library) for "lime" to find the "Install LimeWire.pkg" which is installed in the Library/Receipts folder

Cheers