Will the latter heads work in the patient?
Definitely not.
Or do I need to get another ST3250823AS?
Definitely yes. But not any of ST3250823AS, you have to find ST3250823AS with compatible heads type.
As I wrote
before: If the data is important then you (or data recovery professional probably) have to swap the heads from a donor drive. And it's better to have proper heads replacement tools because of internal parking area (close to spindle motor), these drives does not have an external parking ramp.
Also, will I need to modify the FW
Oh yeah, because we are talking about regular Barracuda 7200.X (not F3). ACE Lab (R) PC-3000 can do this.
Actually, without proper equipment, knowledge and experience I'm totally sure you'll lost your data forever. Bring your drive to a data recovery lab.