If you make Copy Time a universal HotKey, it is very efficient, i.e., the HotKey is active in all apps, so you don't have to have the ExpressScribe window active.
Then it's a two-step process: With your target document active, Copy Time using your HotKey and then Paste the timecode into your document by simply using the paste command, e.g., Ctrl-V.
There's likely a way to combine the two commands into one HotKey that I don't know about...yet.