My rather tiny board leaves to little corner to get the bridges place for manufacturing (at least the one I have in mind).
I do have an idea which should work with a bit more rework after removing the bridges. But that will cut through the 2 inner layers of the board. And I was wondering, if that might cause problems (like accidentally short the 2 layers).
I noticed there is no “fillet” option for zone fill.
Pic shows the desired final outline (fillet) and the necessary contour for the producer
Just manually adjust your fill to avoid the router bit.
Are those dirty white lines supposed to be the outside of the PCB?
In KiCad, the zone fill keeps a defined (in the Board Setup) distance from the outside of the PCB. But to do this, you have to draw the lines on Edge.Cuts, which is a dedicated layer for defining the board edge cuts. (I.e., outside of the PCB.
These lines also have to form some strict rules, the lines you have drawn here do not work, There can be no T-splits, intersecting or overlapping lines on the Edge.Cuts layer.
And you draw it at Edge.Cuts and after zone refilling (hotkey b) you still have copper crossing the board edge?
You can also experiment with Margin layer to see how it works.
After I wrote that, I was thinking the same thing.
-Draw the corners, refill the zones
-change the edge to the extended shape and do NOT redraw the zone fill
That is what you meant, right?
Only thing is, that has to happen after the “final” DRC.
I just suggested that it looks like after correcting corners at Edge.Cuts you forgot to refill zones.
If you refill them they should avoid crossing lines at Edge.Cuts layer so there will be no risk of accidental short between copper being cut without any clearance.
Same thing as what?
I never put the edges of zones on the same place as lines on Edge.Cuts. I always make the zone a lot bigger, and then let Edge.Cuts clip the zone so it stays inside the PCB. In the screenshot below, you can see 4 different zones from different power nets. The slanted sides of the zones makes them easy to spot and select. The zone also keeps the same clearance from the rounded edge.
This also has the advantage, that if anything goes wrong with Edge.Cuts, it will be very obvious when you are inspecting gerbers. For simple rectangular boards, I often draw the GND zone as an irregular pentagon around the PCB for the same reason.
Your first screnshot is confusing. The form / line color it’s not clear on which layer those lines are.
This was the outcome, I was aiming for:
(board cut is needed for production, fill zone should have a fillet, silkscreen line for cutting the final shape)
Can be done, but nit with the fill zone outline. What I did:
Edge.Cut with fillet
Zone refill
change the Edge.Cut layer to the production needs
Downside: Do no DRC after that, because that refills the zones again.
(But I will keep the “fill-zone edges well outside of board cuts” in mind, thanks for the hint)
Don’t design things that way.
To have such PCB shape and zone limited to this arc I would use arc line at Margin layer.
Have you tried how Margin works?
I was thinking this might be wrong, but the only way to get an image to describe the problem.
I’ll search some tutorials on margin layer and be back.
Can this not be achieved with a keep out zone? This would then allow the DRC to still be used.
No need for tutorials, just move the arc in your first screenshot from edge to margin. 
(layer)
Then re-fill zones.
This is a perception I call “annoyingly easy” 
aand thanks.