You can actually edit the published script file (.py). The only thing you need to do is to edit the file in some IDE (I use PyScripter) and re-run the GP task. No restart of service is required (at least when doing those modifications I needed, it wasn't).
However, it is not recommended by Esri and I completely understand why - as soon as you edit the published script, you need to update the source one too. Then you make changes here and there and then you've got two out-of-sync-files which are not equivalent in terms of code, since after publishing you get most of your variables updated by internal Esri variables.
For me, editing of published script file (.py) "just works" on my machine and the development server I work with (no special permissions required). Perhaps in your case, your environment is more restrictive and some adjustment to what and who can edit the arcgisserver folder content is required. Try running a Notepad as an Administrator, it could be UAC permission issue which I have seen once on another test server. Then add the .py file, edit it and try saving.
Make sure the arcgisserver folder is shared (right-click folder > Share tab > Advanced sharing). Your OS account you are logged in should have at least "Change" permission to edit the data in this folder. In addition, share the folder to get it to use the UNC path
\\machine\arcgisserver (right-click folder > Share tab > Share button). Make sure this folder is shared to your OS account. Under the Security tab of the same Properties window, make sure you have Modify privilege and that you have got a permission type Apply to "This folder, subfolders and files".
All of those modifications is something you can set up on your sandbox machine. Show caution if making any permission changes on a production system.