How do I win the mock battle against the USS Republic?

It is possible to beat the USS Republic in the initial battle, as shown in the following video at about 5:50...

There's no real secret to winning the battle that I can find, all of your tactics are correct, but there are a few others that can help in space combat (some of which you already noted, but I'll include them for posterity):

Screenshots are from the GOG.com version of the game, and from the digital official game materials included with that purchase

Prepare for Battle

Immediately hit S to raise your shields and W to arm weapons

Tracking

Use the radar to figure out where your opponent is. The shot below shows the Republic (green dot) at my six o'clock:

Target Analysis

Spock can give you damage analysis on enemy ships, but it hides your own ship diagnostic information while it's active:

Vary Your Speed

Speed up or slow down using the numeric keys as described below, in the official game manual:

Repairing

If you take damage, use the D key to pull up the "Repair" menu and tell Scotty to get to work:

Left to right, these are your shields, phasers, torpedos, sensors, bridge, hull, and engines (I guess warp and impulse are lumped together).

Integrity Monitoring

Monitor your shield and hull integrity using the diagrams on the left and right of the bridge. These can tell you which way to orient your ship so you can keep damaged areas away from enemy fire.