Can you go see a display model of the fridge somewhere? I would go try to take out the shelves and drawers with the door only open that far and see what happens. Getting things out for cleaning would be my biggest concern.

I'm thinking French Door might solve any problems anyway since you'll be able to open both sides, but never having had one, I'm not sure. Seems it would work a lot better than a side by side though.

The spec sheet on the fridge should tell you how much clearance you need on all sides. The salesperson should be able to get you the info or you might be able to get from website.
The KA FD we are considering is 36" and needs 3.75 on the wall side, but perhaps you can get a smaller one?
Also keep in mind that you will want the doors to open wide enough to get drawers out.

So many kitchens have this same arrangement and people manage just fine, but it is one of the things I am looking forward to changing in my kitchen, honestly. jyokoe is right about the specs so definitely pay attention to these. My guess is that the best design for you might be the bottom freezer.

This unit is 21.9 cu ft Architect II, 69"tall x 35 1/8" wide (with door open to 90 degrees) x 33 1/2" deep with handles. KA specifies that you need 1/2" in every dimension plus 1" at the back for air flow, a little more space in back for a water line if you have one, and 2 1/2" on the hinge side if installing next to a wall, for the door to open fully and drawers to pull out.

It'll be fine.

In my last house my fridge was against a wall. I was able to open it enough to open drawers and get anything I wanted. (3 different fridge's over the years) The only time it was a problem was when I wanted to remove the drawers for cleaning. I rolled the fridge out enough past the opening on the wall to remove the drawers.

My father in law has a side by side against the wall and the fridge side doesn't open all the way. If you were getting a fridge like EAM's post, you could do a left hand door. Even a french door fridge I would want to be able to open both doors all the way to get shelves/drawers out for cleaning

There are some fridges that the door opens without needing any space next to it. Even if you don't go with a SubZero, go look at them to see the hinge thing. Others have it too. Just make sure to keep that in mind during your search.

If I couldn't get the drawers out, it might drive me crazy when it was time to clean in there. Unless you can access the area behind the lower drawers by removing a shelf above them.

With a 38" wide space, you could install a 'fridge around 35" and have room to open the door. Sears carries 207 refrigerators in the 35" to 35-7/8" range. That's a big refrigerator and a huge selection - and that's just Sears. Do be sure to check the manufacturer's specs on space needed next to a wall.

Your kitchen is looking awesome! It looks to me like there is space to the right that will allow your refrigerator to be opened enough to be fully functional. I would have the trim carpenter add a fill strip to the right side and a panel to left to make it look built in like the picture below. My refrigerator (electrolux french door) is close to a wall and if I do need to get a drawer all the way out for cleaning I can pull the refrigerator forward to open it fully. Most refrigerators are on wheels and can be easily moved back and forth.

[Traditional Kitchen design[(https://www.houzz.com/photos/traditional-kitchen-ideas-phbr1-bp~t_709~s_2107)

Below is a picture of my Refrigerator installed with filler strip on both sides and also above it to achieve a built in look. The cabinet above it is a 36" wide. Refrigerators only get pushed back to the edge of the door. This leaves at least 3 inches protruding past the front of the cabinetry.

As others have said, check the specs on the possible fridges. I just went and tested mine. It's a KA bottom freezer with a french door top. It has the full-width pull-out deli tray in the fridge. I can pull out that deli tray if the doors are open at 90 degrees -- I don't have to open them past 90 degrees to get that shelf out. I would think most modern fridges are designed this way, so that the hinges pivot and the doors open within the plane of the body of the fridge -- not outside of it.

I think the problem people run into with walls is the wall that the door will be opening against. It becomes a problem if that wall extends past where the door handles are (when the door is open) because then the handles will keep you from opening the doors 90 degrees. But in your case, it looks as though you have an open doorway that the handle will open into, so you shouldn't have that problem.

It will depend on your refrigerator. The different makers have different thicknesses of doors (only the outside part - the doors almost always open to the side of the unit) so the minimum is enough space to allow the door to open to 90 degrees. I think the handle will completely miss the small piece of wall in any event.

Whether you can make the drawers fully work with a 90 degree door depends on the design of the unit and how the interior door shelves are designed. Some need the "gallon on the door" stuff to be able to swing out of the way, while some of the designers seem to have considered that someone may not be able to have the door fully open.

Since you said "planned to buy", you can adjust your ref a bit - maybe even consider a slightly smaller ref - like a 32-34"? I'm just warning you that there are very, very few candidates below a sub-zero price point that have completely usable drawers with 90 degree doors (like none that I know of) so shop carefully.

"It becomes a problem if that wall extends past where the door handles are"

More than just the handles on many of them.
The doors have gotten very thick for insulation to improve efficiency.

Some models are smaller clearance than others.

You just have to check.

Do it like rt did. Get a 36" width unit (or slightly less) and put a 2" filler to the right. That's about all you can do at this stage.

Also, when you go shopping, check very closely how far you have to swing the door to get the drawers open.