Can I monitor 25A peak to peak sinusoidal current wave precisely with this Hall-Effect Sensor?
The quick answer is no.

If you examine the data sheet, you will discover two facts:

1) At +/- 31 amps full scale, the output voltage scale is 45 mv/amp.

2) Output noise is rated to be 11 mV.

This means that the noise component of the output signal is equivalent to about 250 mA, or 5 times greater than your desired accuracy.

Granted, this is a simplistic analysis, and accuracy can be improved by various means, particularly heavy filtering of the signal to exclude as much of the noise as possible. This, of course, will add to the delay which worries you.

However, your concern about delay is completely unfounded, since the figure which you show is not what you think it is. The figure simply shows how long it takes for the output to settle after power is applied to the chip, and since I presume you would be applying power continuously. But since you did not realize this, and were unable to appreciate the figure's caption, I'm dubious that you can compensate for filter delay.

So the longer answer is no, also.