Daniel,
Sorry for the delay in responding. I did see this earlier, and meant to return, but once read, it never again appears unread - and my memory... hmmm, lets just say sometimes there seems to be no film in the camera... <_<
Back to your question...
An Estie inner cap can be removed without a professional cap puller, depending on your mechanical inclinations.
For your problem, shellac or CA may be the simplest solution provided it's a user and the fix isn't being put in the category of 'restoration'.
Removing the inner cap, pressing / drilling out the jewel, and then tightening the rivet holding the clip in place is the best solution.
A fairly crude puller can be made by using a 5/16" - 18 tap that you thread into the inner cap, then you have to fashion some way of pulling the tap and inner cap from the cap itself. Ingenuity helps here, but a simple method might include fitting several washers over the tap shank, fastening the shank in a vice so that the washers have a tiny amount of space available between them, and using a couple of screwdrivers to pry on two sides between the washers to force the cap off. This distributes the load over the whole cap lip so that cap cracking is avoided.
I assembled a screw out contraption using hollow threaded tubes available for lamp wiring, and wing nuts for them. With a screw locking collar on the tap, I just screw the inner cap out. I use 2 washers there also to aoid the rotational force on the cap lip. The best washer to use would be one of those that come with the cap pullers - that has the radial bearings incorporated - but I don't know where to get those.
You could consider drilling the end out of the inner cap so as to reach the bottom of the jewel so as to be able to remove it by pressing / drilling. I suspect that the top of the inner cap isn't particularly essential, but it could be replaced with a plug after the tightening was completed.
If you have access to another jewel, you could drill out the present one and push the inner cap out from the top, or just use the access to the clip/rivet interface to use CA to lock the whole assembly down, then replace the jewel.
Instructions for replacing Estie Jewels are available on Brian Anderson's site
HERE So there you have it. A few ways of getting to the clip fastening itself, and once there, tighten with a punch and anvil arrangement, or glue.
Hope that helps.
Gerry