Bcrypt is bcrypt. You'll need three things to use the bcrypt function: salt, key, and cost.
As long as you can supply the three required values by some means, and the libraries are not broken, then the bcrypt hash result will be the same - parameters and result may need to be converted between byte[] and a hex string or whatnot, but the hash value will be the same.
The salt and the cost is sometimes encoded into the "hash" - such as being concatenated into a single string. In this case, it should just be a simple transformation to create/extract the appropriate parameters and interchange format.
Since in this case the hash exposed and generated externally, I would use an extra round or two than that used for remote authentication - bcrypt with proper round selection is designed to mitigate brute-force/GPU attacks. (Using a different number of rounds will also make a locally brute-forced - but not real - key invalid when applied to the server.)
Also make sure to use a good salt function, such as the hash of a random number from a large domain. As per above, I'd choose a different salt than that used with the remote authentication.
Of course, since this is all client-side, a savvy user could bypass any sort of authentication. The above notes, and I do believe initial choice of bcrypt, are to ensure that the password remains secret - at least insofar the code is able to maintain. (Password reuse is a plague; and for some people, it very well might be their banking password..)