Well then no congrats to you, because you can have a reflector that’s great for throw, but not necessarily hot for flood and vice versa.
Snide comments aside, you could just get something like a BLF GT, which happens to have a rather large flat at the focal plane, and add a ring of LEDs there. And it still probably wouldn’t have as good a flood as if it had a shorter reflector. So no matter what, there’s always some compromise - I won’t call a possible 40-50 degree beam much of a flood, but I suppose it’s all relative. For you, 10 degrees is probably considered flood.
But there probably are more oddball ideas out there… how about making a recoil flashlight, that has a forward facing COB in the center where the aperture is? The SyniosBeam has a large aperture on the reflector, which would fit a nice sized COB. Something scaled down to a more familiar format might be cool?