Let's assume we are working a smooth projective variety. Let $C$ be the category of vector bundles constructed by taking successive extensions of line bundles of the form $\mathcal{O}(n)$ for $n\in \mathbb{Z}$. I have several related questions:

Which vector bundles on $X$ admit a finite resolution by vector bundles in $C$? There might be obstructions in chern classes. You can assume the vector bundle on $X$ has whatever the chern classes you want it to have. For example you can assume all of its chern classes vanish.

Are there examples of varieties that this happens for all vector bundles?

Is it true that for ACM (arithmetically cohen-macaulay) varieties (2) is true?