I went to a honda garage here and they weren't that helpful. Other than clarifying that position lights aren't the same as flashing lights (for some reason I got mixed between both), they confirmed there's no real other way than changing the light.
However they weren't able to confirm if the lights from here would fit the same way with the same cables (are flashing lights and the 3 other types of lights positioned the same way or not). Neither did they know if the frame was exactly the same.
So I am still lost on what to do. I found a place in France where I could order OEM parts for 160 EUR a piece (so 320 EUR total for both headlights). But will that work? Any more advice for me?