Hi Sarah, I imagine these windows could be treated in the same way as conservatory roof blinds with Pleated blinds. These would be available in blackout options. Alternatively, shutters can be made to unusually shaped windows like these and I think they'd make a really stunning feature of such lovely shapes! Here's an example: Enjoy dressing these beautiful windows, Zoe

Blinds that sit within the frame, below the top arch, would be an option - either Roman blinds or wooden slatted blinds. Or if budget allows then Shutters would look fab!

If its rented (so no major changes) have you considered window frosting film? you can get in a huge number of fantastic patterns so you can go wild, maybe if you're on the 3rd floor you could put it on the bottom half of the window so you can still see the sky and look down when you're up close but otherwise not be seen? https://www.amazon.co.uk/window-frosting-film/s?k=window+frosting+film

As you say shutters would be the most expensive so you are down to either blinds or curtains or both. If you are more contemporary with your choices then I would opt for blinds but if you feel as though you want something more fancy with texture then curtains fits the bill. I originally had curtains at my bay window but they were a pain to keep dragging around the corners of the pole so also invested in blinds which have been a godsend. We only use the blinds now at night but the curtains offer a 'frame' on the window as the lounge is on the front of the house so at least it looks as though we have something at the windows for passers by as you cannot see the blinds when they are up.