As the title suggests, am I able to apply 12V to each winding of the 28BYJ (5V version) to increase the torque?
That depends if you mean the ultimate torque or the torque at speed. When a stepper motor is being rapidly switched, the back EMF of switching the windings means that rated supply voltage can't immediately achieve rated current, and if the dwell time on each step is short enough that full current is never achieved, torque will fall off. This can be combatted by using a supply voltage several times the rating with a chopping current control. But the duration of application of full voltage will be quite short, and only as long as needed to achieve rated current.
I understand that it's not ideal as it will overheat, but it won't be running non-stop. It will run on average 2 seconds out of every 12, which I figure will cool down a bit between rotations.
2 seconds is probably too long. The concern is not only thermal but also that some types of magnets that have been used in such motors can be damaged by excessive fields produced by excessive currents.
What you can do is:
1) Source a motor suited to the mechanical need
2) Use a chopping current driver with a supply voltage a few times the rated coil voltage to make sure full current is achieved, and set the current appropriately
3) Optionally have two current setpoints - one for actual movement, and another (possibly zero, ie, motor off) to back off to under software control when full torque isn't required. This isn't unlike your original idea, the difference is that you really shouldn't be running a motor at more than twice its rated current, unless you have an actual spec or informed evaluation that such current is acceptable for brief durations.