If there is not enough room for an inside mount then you can't mount them inside. The only solution is an outside mount (which never quite looks as good as the inside mount). In your situation you could do two outside mount shades but sorry to say, it will never look like your inspiration pictures.

Get a sample blind and hold it up to see how it looks. Maybe you can also consider beefing up the depth of the window with molding surrounding the window, which gives the effect of a deeper window.

Do some on-line shopping for a window blind that you like, then look at the specifications for that blind. It will tell you exactly how much depth you need to mount that blind.

The blind does not have to be fully recessed, it just need enough room for the mounting bracket. For example, for the blind below, fully recessed needs 2 inches, but the minimum recessed depth is only 3/4 inch.

Levolor natural woven shade

Wow those views! I have to admit I would not want to cover them up in any way as lovely as those blinds you have pictured are.

Don't let them tell you that! Woven blinds can stick out a bit if your window is not deep enough, but that really doesn't spoil the look at all. What does matter is if the shade is too long it does get too heavy and will sag in the middle, so I would split that back wall into two shades. It will work with 1" and just protrude 3/4 to 1".

Thank you for the comments that there is still hope!! I will check the depth and cross my fingers that they can work. The guy who measures is the installer and seemed adamant that it could not be done. So hoping these tips give me leverage. If I have to outside mount, I would use two so that it won't be too heavy as mentioned.

So ideally on the back wall I would have 4 shades.

So after a call to the mfg, they can mount with 3/4 inch. They are coming back to measure! Crossing my fingers!!

Closing the loop on this. Got just what I wanted and very pleased!