Are HMACs based on hashes with larger bit-lengths also more secure? When doing encrypt-then-mac, I can choose to use a hmac as the MAC. For example, I could use a hash like SHA-256 or SHA-512 (by using it as a keyed hash) to create that HMAC. 

Does it increase security when choosing a hash with a larger bit-size? Is a HMAC based on SHA-256 weaker than a SHA-512, or doesn’t it really matter because both are cryptographically secure?
HMAC-SHA-256 is sufficient for up to 256 bit security. Confer e.g. NIST SP 800-107. This recommendation is based on the premise that collision attacks are infeasible against common uses of HMAC, and that you consequently only have to worry about primary pre-image attacks that attempt to recover the secret key (and use this for forging subsequent messages). 

SHA-256 has 256 bit security against primary pre-image attacks. This is sufficient for all practical purposes.