Yes it can be done.
I should point out that I only had my LR for about 9 months so far, haven't used it a lot, and still learning my way around, but yes, it can be done as I have found out about it.
Point of note: The geotagging information will become part of the image's metadata, the EXIF thing, or put it other way, if you right-click the image file in Windows, and click on Properties, the geotagging information will be included in that. Therefore, if you want to save geotag information to original files not Library copies, any changes you make to image files in Library, like adding keywords, adding copyright information, etc., they all will be saved along with the geotag information too.
Depending on your Lightroom's set up, in the Library, in Grid view, you may see in the upper right hand corner, a little icon looking like three lines and a downwards arrow, that lets you know the metadata needs to be updated, and if on the lower right hand corner, there's a little icon looking like a signpost, it means the image file now got geotag information. Unless you told your LR you prefer not to display those icons, in which case you'll have to remember which of the photos you had geotagged yourself.
When you know that photo had been geotagged, just right-click on it to bring up sub menu, then hover over 'Metadata' which shows a triangle arrow pointing to the right, where another sub menu will pop up, then click on the one that says 'Save Metadata to File'.
Actually this is a one by one thing, like I pointed out, I'm still new to LR and still learning how to use it, so I haven't yet learnt how to get it done with so many photos.
I think IICR, highlight any photos you want (usual stuff like click first photo, hold down shift, click last photo, or hold down Ctrl and click whichever photo you want to select), then when you got the photos you want, I think you go to the menu bar, click on 'Metadata' to pull it down, then can't remember which one of those commands to use, either the one that says 'Sync Metadata' or most likely the one that says 'Save Metadata to File'.
Yes, you can save the geotag information to the original files on your hard drive.