The only thing I see as a real problem is the recording. The number of phones and lines is just one of scale. You would need a medium to large server farm to support it, but with Asterisk combined with SER, it could be done.
However, the recording is another issue. In any financial institution, recording is a normal thing. (Especially for financial groups that do trading.)
Asterisk can record calls, but asking an Asterisk solution to record 300 calls simultaneously, store the recordings, and have a way to be able to easily retreive the recordings, is asking for a lot. It doesn’t matter much if you have a recording if you can’t find it later when you need it.
The process of archiving the recordings alone is going to make this a real problem to implement. Most financial institutions need recordings to be available for a long period of time (Here in the US, the Securities and Exchange Commision mandates 2 years for recordings of telephone calls involving stock and securities trades.) You could be horribly overbudget just trying to build the collection system, not just the archive solution, but having a way to easily find and playback a recording is going to be essential.
15 years ago, I worked for a bank that lost a recording. It cost them $18,000 to resolve the dispute.
There are lots of commercial recording solutions out there that can handle this volume, but they’re not usually setup to handle VOIP calls. You may find that locating a recording solution that will fill the bank’s needs will be the first problem. Once you find that, you’ll know if you can use it with Asterisk or if you have to build a standard PBX solution.