Charlie,
On the KL1/TID, there isn't enough wall thickness to bore out the TID. It would need to be shortened in length to get it to seat down further. You could drill and tap for new set screws but they will be setting on the threaded bezel cap and the pressure is likely to deform the cap and possibly break the Loc-tite loose? The TID could be re-worked and made to fit but it is not trivial.
On the other hand, the TID does fit your LGI fine! If you remove the rubber grip from the head the TID sits well and the set screws will reside in the grip groove.
The pic below is for Josh, granted the E1E is host to a New Beam but it's still an E1E on the outside
- Don