I often have to duplicate an existing connection just because it has a custom icon.
Wouldn't it be nice to add all custom icons to the icon library so it's much easier to reuse? To remove the icon from the library I would suggest to drag it out (a common Apple pattern).
A little photoshop mockup:
icons that appear in the icon library are actually stored in Royal TSX' app bundle. You can add additional files there and have them show up in the icon library.
The path is: /Applications/Royal TSX.app/Contents/Resources/IconLibrary
If you do so, you have to ensure that the icons are provided as a 1x variant with 16x16 pixels and a 2x variant with 32x32 pixels. They also have to be of .png type. And if you move between multiple Macs you will have to keep icons in sync yourself.