We know there is only one reality, measured from different reference points. So the pertinent question is what is the reality in this case? Does the particle cross the event horizon or does it not?
I don’t believe it does and here is why: The equation you set out is the Schwarzschild metric which describes the space/time around a point mass. This is also used to model a stationary black hole with an event horizon. As a particle falls radially towards the event horizon, time progresses differently depending upon the reference frame from which it is measured, because of the effect that gravity and motion has on the passage of time. This is all reflected in the Schwarzschild metric
A first integral equation that is derived from the Schwarzschild metric allows calculation of the passage of coordinate time, as experienced by the distant observer. A second integral equation, also derived from the Schwarzschild metric allows calculation of the passage of local time, as experienced by the particle. As long as the particle is outside the event horizon of the black hole, both the integrands are defined and the integrals are perfectly well-behaved. So the journey of the particle can be tracked with certainty all the way until the event horizon.
As the falling particle gets closer to the event horizon, time goes quicker when measured in local time than the journey measured in coordinate time, because of the relativistic effects of motion and gravity.
Since velocity equals distance divided by time, there is a different perception of speed depending upon whether the distant observer or the particle is making measurements. For each location reached, the particle thinks it got there quickly, so it thinks it is going fast. The distant observer thinks the particle got there slower, so it thinks the particle is slowing down. The particle arrives at the same location, there is just a different perception of the amount of time it took to get to the location.
This perception of the speed slowing down from the perspective of the distant observer continues until the forward progress of the particle seems to be very slow. So slow, in fact, that at 10^60 years or so, when the black hole has completely evaporated, the particle is still outside the event horizon. The particle’s journey stops at an end location that is near but outside the location of the event horizon of the no longer existing black hole.
Now, from the particle’s perception, it was going pretty fast when it reached the end location. It was trucking along at a fine speed, when all of the sudden, in the local time of the particle, the black hole rapidly evaporated.
This is the scenario from the calculations of the integrals derived from the Schwarzschild metric. As long as the particle is outside the event horizon, the integrals used to calculate the journey are perfectly behaved, so there is no need to doubt the results. Further, according to Einstein, all coordinate systems (reference frames) all will observe the same reality (e.g., logical sequencing of events), although the time to complete the journey will vary depending upon from which reference point the calculations are made.
Given the above scenario, it seems that regardless of the start time, nothing is able to cross the event horizon.