Imagine a scenario where a client application is sending a password to a backend server so that the server can validate that the user entered the correct password when being compared to a stored variation of the password.
The transport mechanism is HTTPS with the server providing HSTS & HPKP to the user agent and strong cryptographic ciphers being preferred by the server scoring A+ on SSL labs test. None the less, we may wish to avoid sending the original user provided password to the server from the user agent. Instead perhaps we'd send a hash after a number of rounds of SHA-256 on the client.
On the server-side, for the storage of passwords we are using bcrypt with a large number of rounds.
From a cryptographic point of view, is there any disadvantage to performing bcrypt on the already sha-256 hashed value as opposed to directly on the plain text password? Does the fixed length nature of the input text when using hashes somehow undermine the strengths of the algorithm.
EDIT: I'm not asking about performance such as the memory, CPU, storage requirements or wall clock time required to calculate, store, sent or compare values. I'm purely interested in whether applying a hash prior to applying bcrypt could weaken the strength of bcrypt in the case of a disclosure of the full list of stored values.
For anyone interested in this, I followed advice and asked on security.stackexchange.com here