Are you trying to collect images from several different folders on your computer for some kind of upload somewhere else? Like, say you're updating your website and you are needing the actual files to upload, and you're trying to organize in Lightroom, but then be able to upload the files themselves from one common folder on your computer instead of tracking down 30 photos from 25 different folders?
If so, the issue you'll run into is that Lightroom doesn't let you have multiple copies of the same image unless you're importing them and allow duplicates to be imported. It won't offer you an option to make a duplicate within the program itself, that I'm aware of, since that's sort of antithetical to it's function as a database, not explicitly a file manager. You can easily do this in Bridge, for example, or Windows Explorer.
But if you're using Lightroom to organize, then two things come to mind.
1) Take that smart collection in Lightroom of all your images from various places, open Library Grid view, select all, and then drag and drop the images to a specific folder in your Folders panel. The problem is that this will move all the actual files to one common folder on your computer, not make a copy, and they won't exist in their different folders anymore. Maybe that's ok for you?
2) The other way would be the Export, as you're saying. You should be able to export them to one common folder using the export dialogue, and that would be your best bet, I think. Using the Original setting for Image Format in the File Settings section will strictly make a copy, with no adjustments you might have made to the files in Lightroom. You might not have made any anyway, if you're only using it to organize. But if you did make edits you want to keep, and you don't want to change the file exptensions, then I would filter by file type in the Grid view first, export all the JPGs, for example, using whatever export settings you'd like, then repeat the process for the HEIC files. That will allow you to keep their original file formats, apply whatever edits you've made, and have updated copies in the new folder. Or you could just export all of them to one common file format - it wouldn't have to be JPG, as you can select various options in the same Image Format dropdown and keep the edits that way, too.

Barring those, you could go into Bridge or Windows Explorer and manually copy the files you want that way.