The dual-axis controller on my EQ5 (looks like
this) has a microcontroller (with firmware). It reads the button inputs, calculates the tracking rates from it and controls the stepper motors.
I cannot find the link but I read about someone building their own EQ5 controller using a PIC microcontroller and writing their own software. That version could run at 16x as well.
Then I started building one myself and I did some tests - it could do about 24x before motors started stalling. It turns out that the limit was imposed by the combination of a cheap stepper motor and a gearbox with 100x gearing ratio which was needed to achieve the resolution.
GOTO drives use higher quality motors with microstepping to achieve the resolution instead of high gearing. If you look
here and scroll down to the two images you will see the difference in gears used in HEQ5 and HEQ5 Pro (with GOTO).
Anyway, back to your original question, it would be difficult to reprogram the original controller. Building/buying one for your mount and which has programming for 16x would possibly work. It worked on my EQ5.