When to use an anti-backlash bolt for a Z drive gear?
First of all, let's look at the geometry: We have a static XY and changing Z on our build plate. This mitigates several problems we might have faced in a different setup, mainly that our lead screw has loads changing from one face to the other: all loads always go into $-Z$, so the lower surface of the Trapezoidal threads in the brass nut of the bed carrier and the upper surface of the screw will always rub against each other.

However, let's talk about your lead. Tr8x8(p2) is not a Metric ISO-Trapezoidal pattern that is commonly recognized - the only 8 mm diameter one in that standard is Tr8x1.5, very flat and very well suited for putting high loads onto it because there are several engagement surfaces in the nut and a low pitch angle - the thread only has about a 5° (+-1°)to the horizontal. Assuming a 15 mm nut, that's 10 times the projected engagement surface for a total area of about 214 mm².

A Tr8x8 is a very common oddball as in, it's not in the metric tables but available everywhere. It is considerably steeper, and with a 16 mm nut (for easier calculation) would result in 2 times the projected surfaceof a single engagement per thread. But that's only a projected area of 42.8 mm² per thread - about 1/5th of the Tr8x1.5, which correlates directly into that much less friction, because the friction is linear with the surface, which is linear to the length of the nut.

Taking an "intermediary" TR8x4 gets us about 1/2.5th of the Tr8.1.5's holding power per thread at the cost of half of the speed. From a physical standpoint, I'd  take this one, also increasing the precision of the Z-movement by a factor of 2 in comparison to the Tr8x8.

Tradeoffs

Generally speaking, we have a tradeoff here:


Pitch is proportional to the movement speed


double the pitch and you move twice as fast

Pitch is anti-proportional to holding power/torque and Z-position accuracy


half the pitch gives twice the torque and the Z-position error is in average reduced by half

holding power is proportional to the diameter of the rod and the number of threads engaging with the nut
Z-position accuracy is anti-proportional to the effective step-size of the motor 


Z-position accuracy can be increased by using a reduction gearset: a 2:1 reduction gear halves the step-size and doubles the torque at the price of halving the movement speed. However, it can and will introduce slop and backlash.



Due to the geometry, the slop in the nut should not be an issue, as the relevant engagement surfaces never change. Further, the heavier the bed, the fewer problems you'll have with fast movements, as gravity will prevent the bed from going "airborne" at the end of travel towards the printhead. Travel away from the printhead, as it will do during printing, should never experience such.