Well, it would probably help if you shared this information somewhere on YOUR site and not after lengthy discussion here. You've not even removed the outdated warning about Eagle parts with multiple gates yet. Anyway, I'm afraid there is no "clean" way to resolve this, or at least not in fully automated way:
1. I named my part SR10010F-0204 because the rest of the part number is mostly irrelevant from the schematic and footprint standpoint. Flat, slotted and round shafts of given length will look exactly the same on the PCB, for example. So under present SnapEDA system, I should upload the same library 3 times, only with different part numbers. That doesn't look very efficient to me, because many electro-mechanical parts are like this.
2. On the other hand, if you allow any parts to be public, then your site will likely become swamped with lots of libraries with no clear connection to real components. That's probably even worse.
Where do you get the list of existing part numbers? I mean, the SR10010F switch is a typical example of a part that can exist in literally dozens of configurations. Just because e-shops like Mouser offer only some of them doesn't mean the other configurations don't exist.
Moreover, there are many "exotic" components which you won't even find in such e-shops, because the manufacturers sell them directly. For example, I originally planned to upload library for Compulab CM-FX6 series of processor modules, but they're not even listed on SnapEDA. But at the same time, libraries for these "exotic" components are usually the most laborious to make, so sharing them with others has the biggest "value" for the community. In other words, there should be some way that users can add new part numbers. Or maybe not directly add them, but suggest them to a human moderator, along with proof of their existence (datasheet). He/she would then add the part numbers, maybe including all possible/available combinations. Do you employ somebody who could do this?
https://www.compulab.com/products/computer-on-modules/cm-fx6/Edit: the upload indeed works when I changed the name of my library to SR10010F-0204-20F0B-C7-N-0027.