Does Identity-Based Encryption actually solve any problem?
IBE is advantageous over standard asymmetric methods in one aspect, and that doesn't appear to apply in the case you're interested in.

In both cases, IBE and asymmetric methods require an enrollment process (whether to distribute secrets, or authentication data), so there's no real difference there.

However, when Alice wants to send a message to Bob, with standard asymmetric cryptography, Alice and Bob needs to exchange messages (whether to perform a DH exchange, or for Bob to send Alice his certificate).  In contrast, with IBE, Alice just encrypts the message with Bob's public id, and sends that message to Bob; Bob doesn't need to send a message to Alice.

So, that is the advantage of IBE; if you are sending unidirectional messages, you don't need a back channel to complete the exchange.  If this is important for you (for example, if you are sending encrypted emails; the store-and-forward email architecture doesn't allow a back channel), this is quite useful.  If you do have a convenient back channel (as in your case; you're going to exchange messages encrypted with a symmetric key anyways), it doesn't gain you anything.