Thank you. That helps a lot.
I think I now better understand the challenge now that I can compare the orientation with your print result.
The short answer is that I don’t think you can get both sides to create a clean print as if you were to have them flush. Since neither side is flush, you obviously do not have that option.
So here is one method that might work but there is a leap of faith as this may seem counter intuitive.
If I look at your print, what I see is very shallow supports. They are a bitch to get off clean as you no doubt have shown in your image.
So here comes the counterintuitive part. Make the supports taller and they will snap off away cleaner.
There are two parts to this solution and you can try them separately or together. I might suggest cloning the part and trying them three ways.
Levitate the part above the print bed by using the assembly feature.
You know that the slicer won’t allow for you to just pick up a part, it will just fall back to the plate. So here is how we trick it.
-
Create a small primitive object. I usually use a cube primitive and resize it to 5x5x1. That makes it easier to pick off the plate with your fingernail after the print. for my example, I just so happen to have a part similar to yours on my build plate right now. How ironic. 
-
Now select both parts and make it an assembly.
-
Now click on the object in the object menu( It’s easier than trying to click on it while it’s on the plate) and select the up arrow(blue) and drag it to how high you want the supports to be. It will now stay suspended above the plate because the slicer thinks it’s part of an assembly and that 5x5x1 square is the lowest part of the assembly thus anchoring the whole thing down.
-
Make sure supports are selected under the supports menu and click slice. This will fool the slicer into thinking that this part of the assembly needs support. The square is just a throw-away part. Now that you have a taller support, it may be much easier to pull them off with a snap rather than trying to pry them off. You may have to experiment with taller or more denser supports to get the right mix.
The down-side of this approach is that you waste more filament and of course it can double or even triple the print time.
Change the variations on the support width.
This image is the sliced image but note that I have check off the check boxes for everything else other than supports. This is so that we can see what the affect is of changing supports. Here is the default.
Now if I change the interface spacing to 5 mm, you can see that the supporting filaments become much farther apart. It’s a lot easier to remove wide supports than narrow ones but this will come at the cost of potential drooping in between the lines depending on your filament and temps. But you can use smaller amounts too such as 3mm and experiment. That’s why I suggested cloning the model and printing multiple examples.
You can also increase the maximum interface layers to 3 which may make it easier to snap off. In that example, you are relying on the brittleness to help you.
One last tip.
Before you try to snap off the supports, stick the entire plate in the freezer for 5-10 minutes. This will make the supports much more brittle and make it easier to snap off.
