Yes, it's possible to "upload" a .gs file to an existing Google Apps Script project. To do this you might use CLASP, GAS GitHub Assistant (a Chrome extension) or directly the Google Apps Script API.
Please bear in mind that you also could alleviate these pains (having to update the bounded to spreadsheet GAS project copies) by taking a different approach. You could use libraries or to create a Workspace Editor add-on.
Libraries help by keeping a big part of your code in a single place but you might still have to update the copies i.e. if you are using simple triggers like onOpen
and onEdit
or if you change library function name, add a new function, etc.
Add-ons help by keeping all the code in a single place bu you have to publish the add-on to the Google Workspace Marketplace. If you have a Google Workspace account the add-on can be published for internal use.