To get the SIM card out, the easiest (and I use that term lightly) method would be in disassembling the device.
After you remove the back glass and separate the midframe from the LCD frame (with the motherboard), you'll have to get very pointy tipped tweezers and slowly work the SIM card out of the SIM tray mounted on the motherboard. This is the method that may result in the least amount of damage. If any of the pins on the SIM tray get damaged, you'll have to find someone with soldering experience to replace it.
Here is ifixit's guide to removing the motherboard. (I personally prefer high quality playing cards or thin guitar picks for removing the back glass, with the aid of a razor blade to open that initial gap. Just be very careful and maybe order a back glass just to be safe ([product|IF302-072|Back Glass])
[guide|47476|Galaxy S6 Edge Motherboard Removal]
The SIM tray can easily be found in the ifixit store for a good price.
[product|IF302-066|SIM Tray for Galaxy S6 Edge]