Like you said there are two ways to get there however booking a
transfer service to
Dalaman is not going to cut it. Gocek is past Dalaman on the way to fethiye so your options are to either catch a bus to Fethiye and get off at Gocek turn off or book a transfer to (or taxi) to gocek.
Why dont you Visit us at
ICR office during your stay and we will see if we can save you some money and time. There may be an empty service bus going to Dalaman for a pick up so maybe we could arrange our driver for an earlier leave and he can take you to Gocek first and then go back to Dalaman for his scheduled pick up.. (not a promise but pay us a visit and we will do what we can

Rami