Not necessarily. The Navy redesignates at their discretion. When I was a student, all DORs and attrites redesignated, the vast majority as SWOs.
When I was an instructor, I saw guys who graduated from the academy/ROTC in the spring decide to DOR and they were on the streets with less than a year of active duty.
It changes all the time and if you pick up a specific designator for OCS and are found to be physically nonqualified, it is a lot easier to just do an entry level separation and be done with someone.
To the OP, just pay for it at a testing center. I was active duty and rather than deal with the red tape to get the test done on base, I just went to my local FBO. Much mo bettah.