It all depends on site with the drop down. I have had similar issues where it works or does not. On those that don’t I have had luck using Irfanview. It’s a fee program that will let you set up a hot key to do a screen capture. The I would just edit the saved image to get what I need. You can also capture a cropped section with Irfanview as well.
FOUND A SOLUTION!!
When using the snipping tool, in the middle of the tool bar is a DELAY option. Select 5 seconds, then select new. You can now have 5 seconds to click on the drop down to open it up before the screen capture by area or how ever you have it set works.