You would probably want to use PBKDF2 as a key-stretching function rather than just naive SHA256. Otherwise you're clipping your bcrypt input from "56 arbitrary bytes" down to "56 hexadecimal characters".
I haven't looked deeply at this, but using "key stretching" that clips your output characters to such a small space smells very suspect to me.
Remember: there is only 32 bytes of actual output there, regardless of whether you represent it as hex or binary. And since bcrypt can't take more than 56 bytes of input, you are clipping that down to the equivalent of 23 bytes.
I haven't looked deeply at this, but using "key stretching" that clips your output characters to such a small space smells very suspect to me.
Remember: there is only 32 bytes of actual output there, regardless of whether you represent it as hex or binary. And since bcrypt can't take more than 56 bytes of input, you are clipping that down to the equivalent of 23 bytes.