I may try answering by asking another question: in Earth referrence frame, will the clock ever cross the event horizon?
Now, as you mentioned time dilation would diverge as the clock approaches the event horizon, therefore I would say that in Earth reference frame the clock will never be inside the black hole.
This makes me think that you cannot compare the time measured in Earth reference frame with the one measured in the clock reference frame once the clock is inside the black hole; in fact I think you cannot compare the time measured in any inertial reference frame that is outside the black hole, because of the time dilation you mentioned.
To me the only way your question may have a (very speculative) point is making the clock exist directly inside the black hole, being there since the origin of everything without having crossed the event horizon in order to be inside the black hole.
The answer to this question about whether M87 black hole is an actual black hole might be interesting to you, as the example there made can be compared and used to your question: time dilation does not imply that the clock never crosses the event horizon, but there is no way to communicate with said clock.
In fact, Earth reference frame would receive signals the clock reference frame sent before crossing the event horizon, and those signals would take up to infinite time to reach Earth reference frame.
On the other side, Earth reference frame would not be able to send signals to the clock reference frame once the clock crosses the event horizon and viceversa.
According to this, then, a confrontation between the elapsed time in those two reference frame is not possible