The mounting holes are not connectors, so that process will not work. The way to do what you want (starting from the process you are using), is to export the pcb image from the editor, use a separate tool (like Inkscape) to delete the mounting hole circles from the image, then import the updated svg file back into the view in the editor. That should work fine, as long as you do not change anything else in the file. Inkscape (and some other svg image editing tools) have been know to add custom content to the images that can confuse Fritzing on occasion.
The safest way is also the hardest. An svg image is really xml, which is plain text. A simple text editor can be used to remove the circles, if you can locate them. A text editor has no way to ‘point’ at the circles to locate them in the file. There are various ways around that, all needed some knowledge of how svg images are structured.
Try whatever svg image editing tool you have available. If it works, you are ready. If not you can go back to the original part, and try something else. It is a simple fix, so if nothing wants to work, posting the part file here will likely find someone who will do the delete, and post an update part file for you. Or just the svg image, though can sometime be harder to post than whole part. That looks big enough (part dimensions) for the forum to accept as an upload.