I'm the developer of Greenshot. As there are many channels on the internet I cannot handle all, so I'm a bit late to answer your question. In fact I'm not so active on reddit.
I can tell you Greenshot, as it's available from our website https://getgreenshot.org, currently doesn't have anything that can do this. It's open source on https://github.com/greenshot/greenshot, and all people, due to the technical nature mainly developers, can see exactly what it does and doesn't.
Truth must be told, in theory someone could add something to Greenshot that makes such a feature possible, and roll this out on your PC. But that would be a lot of work, for something which in theory is already possible without Greenshot with less work.
I think your company just want to help their employees with being more productive, and went with an Open Source application BECAUSE it's completely transparent in what it does and there is nothing like that build in. Having multiple screenshot tools in fact is not helpful, maybe they are phasing the other one out?
this has bugged me as well.
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https://getgreenshot.org/faq/how-remove-plugins-or-destinations-from-greenshot/
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tl;dr - edit your ini file while greenshot is not running, go to the ExcludeDestinations line, and _add_ items you want to *remove*, separated by commas.
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for future reference and to add some google search keywords, this is how you remove (or hide) items from the screenshot menu, or how you customize the screenshot menu in greenshot.