If you don't want to use the session in the way it is supposed to be used, then don't use it at all - develop your own session object. It can still implement HttpSession, and even extend from an implementation of HttpSession.
You can use a Filter to wrap your request so that it returns your session object rather than the standard one. In your session you can store things in DB instead of in-memory.
Instead of writing to the DB, you can use Hazelcast - it provides distributed collections. But I guess it will take the same amount of effort as to configure session replication. And session replication is not something that hard - it is supported by all containers.
These are rough guidelines, the task will not be trivial. And I'd recommend sticking to the standard session usage patterns, and storing things in DB only if really needed.
In order to avoid the need of replication you might try using sticky sessions - i.e. when a user is directed to a server by the load balancer, each subsequent request by that user is sent to the same server.
