Is it possible to handle the ssl connection with IO::Socket::SSL directly but still have all the LWP::UserAgent functionality for my interaction with the server after the connection is established?

No, LWP wants to have full control over the socket and does not allow to create a connection using an already established socket.

Well, yes it is possible -- see Re: LWP is there any way to get "real" outgoing headers? -- but it doesn't make much sense .... if there is a bug in LWP suite, fix the bug, not work around it