You could, in effect, "furr out" the interior trim/mullion details on the bow to give yourself more room. A decent trim carpenter or DIY'er could do that.

That's what I was thinking. I will have to see if doing this will void my warranty if I used a nail gun

You can even use some azek trim to furr it out.

You actually have a few good solutions without needing to add any additional wood trim. We come across this problem on a regular basis and it really depends on your preference for window coverings.

Yes, you can simply mount the Top Down / Bottom Up Honeycomb blinds directly to the ceiling. These products have a small headrail and can butt up to each other very closely without leaving large gaps between the window coverings. Photo attached.

Alternatively, you can also use shutters in a bay window and have the top and bottom portions operate separately. That way you can leave the top portion of the shutter open while closing the bottom half for the pets. The divider rail can be positioned lower in the shutter if desired.

Wow. That last picture is really pretty and done very well.