a multilevel deck would probably mean losing some patio or grass space, but would create a larger entertaining area.

Lower the deck to 30" above ground, to eliminate the need for the railing altogether, putting the fence between you and your arms-length neighbor. Broad steps from the house will work as a broad bench on the deck.

Cut the support posts in half and bolt the top and bottom side by side, moving away from the neighbor 4", when rejoined below.

Use the railing as a cute little fence to lean on or set drinks on at some other spot in the yard, or to screen the A/C units.

If you want to maintain the current footprint, you should see if you can move the stairs so that they are to the left of the doors. It would be much better if you could have access to the entire deck just by opening the doors, instead of having to walk away from the door to your small deck. Move the grill area to the below smaller patio and put your dining space on deck. As for a pergola, I'm a fan. With curtains and a cover, you can control the sun - assuming you get a lot of sun here.