I wouldn’t start cutting anything just yet. First you need to visualize as much of the inside door panel and door opening mechanism as you can. That would include raising the window as far up as it can go, removing the plastic splash cover, etc. There are usually multiple connections on the door handle locking button. One to the door lock button, one to the interior handle, one to the lock mechanism. You may have pushed on a second rod that operated the handle whereas a different rod of the mechanism fell off or disconnected.
You might have to hold the window at the top of the frame, use duct tape or something and then lower the window frame itself or the window regulator unit. That should allow you an unobstructed access to the handles and the linkage assemblies.
Once you do that follow the rods from the inside part of the door handle and see that everything is hooked into it’s respective slot or keeper. My guess is you’ll find one has popped out, probably the one that’s the linkage you moved downward to open the door. It’s probably accessible from an access hole in the existing sheet metal of the door. May be covered by something like splash cover or something else. You should also gently tug on the connecting rods and make sure they’re all connected. Sometimes too, the attachment point on the inside of the handle breaks for one reason or another, rust, maybe corrosion or just metal fatigue, the rod falls down into the door interior.
If you can find a copy of the service manual on this Buick, check the diagrams in the Chasis/body sections and you’ll see how it all links up and where. You shouldn’t need to cut anything to get to where you need to reattach something.
One last thing, run a magnet or visualize the inside bottom of the door panel and fish around for loose hardware like nuts, fasteners, springs, etc. I don’t recall any springs involved here except in the handles themselves, but you never know what you might find down there.
Sparks