I have had to do this a couple of times, most recently today when a handle I put on after the new paint job, slipped the lever pivot rivet and quit working.
I have found the whole thing to be a bad experience in general due to the design, but think I have found a way to to it without damaging stuff. To do a decent paint job, you have to remove the handles and remove the rivets in them to get the chrome section off on the Buicks, so you can paint the buckets.
The key to it all probably could be said to have less to do with the handle, but with the other stuff around it in the no room or access door area. Selectively positioning some of the stuff can let you just sneak by without doing carnage to things, most of the time, I think.
If you just try to remove the bucket and handle assembly by taking off the two inside the door nuts, you can't get the handle innards out through the opening enough to remove the clip without bending stuff. If you try to disconnect the link at the latch assembly, it often won't come out readily without excessive force, and if you do get it out, you can't get it back in later.
What I finally settled on, which worked pretty well on both sides of the car, for the front doors.
* Remove the inside door panel an control panels and fold back the plastic sheet out of the way
* Remove the clip for the outside key lock from inside the door from the inside of the door, slides off pretty easily
* Pull the door lock out of the outside until it can rotate if you want to turn it, which you will want to do. Don't remove any clips or links.
* Remove the two bolts for the crossbar slide for the window track that goes right across the access hole and swing it as far up as it will go for more room to work.
* Remove the single bolt at the bottom of the window track, pull the rubber from the track up a ways and push it toward the outside of door to get more room to work.
* Peek into the cavity and see the door latch link connection to the latch which will be the closest to you outboard one. Right behind it is another link which goes to the key lock and is so close to the handle connection that it keeps you from pushing out the link from the handle in the latch.
* Reach around and rotate the entire key lock so the link at the latch moves out of the way of the door handle link.
* Push out the door handle link from the latch, but you need to use a small fork tool as it will be tight.
* Remove the two nuts that hold the bucket and handle to the door.
* Lift out the handle, bucket and link together while positioning to the key lock out of the way for clearance. If the lock is in place you can't get the link past it or back in past it.
* Reverse procedure going back in. Manipulating the key cylinder and link is the critical part with this or you won't be able to get the handle link into the latch connection. Moving the window track and cross bar slide for more space may not be totally necessary, but only takes a minute and makes it much easier.
If anyone has come up with and easier way, I would be glad to hear it, as I fear these will eventually have to come out again so I can make new rivets for the handles if they work loose again.