I'm not sure what your budget is, but I really like the look of shutters on sliding glass doors. It can get expensive, but it could be a good option.

Sliding Glass Door Interior Shutters · More Info

This is the photo of the door. I thought I'd attached it to the original post but I don't see it.

I had a huge glass door in my old living room this is what I did to get rid of vertical blinds. At night closed the curtains. Opened on the right to open the door the go to the back patio.

you could always put a bracket in the middle to hold the rod. but, there are other alternatives,,,,

if the shutters work with the aesthetic of your home/room, they are beautiful

How can you sleep in a room with vertical blinds? Ha! Sorry, just kidding. You should rethink your idea of opening your drapes from one side. Having drapes stacked to one side when the drapes are open looks weird & unbalanced. Plus if you go with less than 2X the fullness because you're afraid there will be too much stack-back on the one side, the drapes will look skimpy when they are drawn. So your drapes will look weird open, and look weird closed. I don't think you will be happy with the result. You should either relocate the mirror and install centre drawn drapes or go with blinds or shutters.

Depending on the condition of the vertical blinds, you could potentially remove the slats and put curtains up instead: https://www.instructables.com/id/Easy-Way-to-Replace-Vertical-Blinds-With-Curtains/

You could try the middle bracket off center closer to the non-opening end. You could put some panels on either side. When you want them open, you could leave them blocking the far end a bit. Or have a curtain holder and swag them a bit. Depends on your needs and style. Good luck!