A bullet of mass 0.045kg is fired from a gun of mass 9kg, the bullet moving with an initial velocity of 200m/s. Find the initial backward velocity of the gun.

In this case, we will use the conservation of momentum.

The forward momentum of the bullet equals the backward momentum of the gun.

m = mass of bullet
M = mass of gun
v = velocity of bullet
V = velocity of gun

=> MV = mv
=> V*(9) = (0.045)*(200)
=> V*(9) = 9
=> V = 1 m/s

The only issue I have with this is the principle of conservation of momentum is that final momentum is equal to initial momentum

initial=0
final= sum bullet and gun
0=MV+mv
MV=-mv
and this leads to
V=-1m/s, and of course, the velocity sign means the opposite direction of the bullet. Momentum is a vector quantity.

Right, agreed.

Though to prevent the need for the negative sign I specified 'forward' and 'backward' velocities, so that I could omit the direction from the calculations.

I know what you did, but your statement is not true: he forward momentum of the bullet equals the backward momentum of the gun.

The magnitude of is equal to . When teaching, it is usually to let the math work out the answer from the math, not from a mental construct.
Otherwise, good. Over many years, I have learned many pitfalls for students to fall into, and how to avoid them.