BCrypt is considered more secure
The theoretical security of bcrypt has received less scrutiny than that of PBKDF2, SHA2 and HMAC. PBKDF2 is thus widely standardised (e.g. in NIST SP800-132 and PKCS #5) while bcrypt is not.
In practice the security (resistance to brute force attack or dictionary attack) of bcrypt and PBKDF2-HMAC-SHA512 can be controlled by a workload parameter. bcrypt is more secure than PBKDF2 in the sense that: if you select the workload parameters such that the algorithms use the same computing time on a general-purpose CPU, then an attacker able to choose their hardware (CPU, GPU, FPGA, or ASIC) can attack PBKDF2 4 to 6 times more effectively than bcrypt at the same cost (see the scrypt paper).
As a heuristic we could say that a password hashed with PBKDF2 should have 2 to 3 bits more randomness than a password hashed with bcrypt to achieve equivalent security (against a brute force attack; for equivalent workload parameters).
Would it be secure to take the sha-512 hash of the key produced by BCrypt and split that
No, but you're on the right path by splitting the operation into a step that produces a key, and a step that expands the key. The trick is to select the right primitives for these operations.
To expand a key - that is, to produce using one key a stream of derived random output suitable for use as key material - you need a Pseudorandom Function Family (PRF). SHA-512 is not a PRF, but HMAC-SHA512 is. Using HMAC-SHA512 would be a secure - but not ideal - solution.
A better solution is to use HKDF which splits key derivation into extraction and expansion steps. By using bcrypt or PBKDF2 you have already extracted a pseudorandom key (HKDF calls this a PRK) from the password. You can apply the HKDF expansion step to that PRK to produce derived keying material of any length; equivalently you could use one of the Key Derivation Functions from NIST SP800-108.
See also Do any security experts recommend bcrypt for password storage? for more on PBKDF2 vs bcrypt.