What is the maximum bullet weight for 223/5.56 that will be stable with a 1:9 twist barrel?
Not every 1/9 twist barrel is the same (might be a 1/9.5 or 1/8.7), a the only way to really know if your barrel will handle the upper limit of normal is to shoot it.
What are you looking as shooting?
OTOH, lighter bullets under 62 grains might give erratic performance, and the lighter ones will tend to disintegrate before reaching the target.
Be well.
Just be careful with copper rounds, like Barnes TSX (a premium, quality hunting bullet I've used). Copper is lighter, and hence the rounds are longer than their lead equivents. Which is why they recommend a 1 in 8 twist for a 70 grain TSX. Many of the bullets mention what kind of twist they recommend (for reloaders anyway).
I don't know why this caught my eye tonight, but it gave me the chance to do a little research. For one thing, I've always wondered why .223 barrels weren't all 1:7. I found some retired engineer (who used to design nukes) that did a calc on how fast a copper jacketed lead bullet could spin without coming apart. Turns out that at a twist rate of 1:7, the velocity limit is 2989 fps. (1:8 = 3416, 1:9 = 3843, 1:10 = 4270, 1:11 = 5124)