Not sure if it does not have variable, however there might be creative solution.
From top of my head...
e.g. depending on your flow and needs, in your action you could specify static name (static.pdf) and location of PDF to be inserted.
Each time you need to insert different PDF, you take that different PDF and rename to static.pdf and move to that location, than run action to add page from that file.
I think it's doable this way, did not try it, but it might help you think in different direction.
that was exactly what i did, the problem is the action list embeds the original static.pdf you select and does not replace it with the new one you submit and replace via Switch client.