Solution
Actually, you can follow the technique that being used in new ticket form in staff-side. There you can see that the default one is only displaying the Department (without Help Topic selection).
You can improve it by adding Help Topic selection in new ticket form at the staff-side, afterwards, implement it at the client-side (Department and Help Topic selection).
You can see the modification code that displaying Help Topic selection at the staff-side that I created based on the similar modification made by someone on this forum:
(Displaying Help Topic in New Ticket Form for Staff in osTicket v1.6 RC5).
That modification will give you the ability to automatically select Department based on the Help Topic that you choose that related to the Department.
Hopefully this can help you.
Sincerely,
Masino Sinaga