Connect two Copy to Clipboard Output, but add a Delay Utility between them (even 0.5 seconds worked, in my tests). On the second Copy to Clipboard Output, have {clipboard:1} instead of {query}.
The simple answer is "use a snippet if possible" because Alfred will do that for you.
More generally, you need to save the current contents of the clipboard (using, e.g., pbpaste), run your Paste action, wait for a fraction of a second for the paste to complete, and then restore the clipboard (pbcopy).
The problem is that pbcopy and pbpaste only understand text. If you want it to work with all types of data, you'll have to mess around with AppKit and NSPasteboard.