As I said, the gravity profile isn't what you'd expect.
Earth has a column of material under each point that is treated as (on average) 6370(ish) km away.
Birch has a tiny column of material very near followed by empty space followed by another tiny column of material at 0.96 LY away. Gravity directly towards the centre is thus tiny at any point. How the vectors for every point add up to act on any other point I don't know but it isn't the same as a solid body. 50% of the mass is over 0.48LY away and is acting at an angle. Nearby mass is mostly pulling sideways and doesn't exactly offset the far points because of the variation in distance.
Spinning the surface at something near C means that any incoming particles are incoming at near C. That makes them very dangerous to life!
The circumference is almost exactly 3 light years. Which means that centrifugal force experienced is about 0.5m/s2 (relativistic effects not calculated).
A sphere will be in trouble because the poles have no centrifugal force and thus fall towards the centre. Here they quickly exceed the mass required for a black hole to form. And the whole thing then collapses.
The great circle has the mass of 1,523,909,050,200.7/2 suns spread along a line 3LY long. That is .000027 MSol per metre. More than enough to make any material malleable. So the radius slumps, collapses and...big black hole.
