Instead of resorting to full drift alignment (it takes a fair amount of time for the object to drift enough), you may still be able to use ASIAIR Polar Alignment even if you cannot see the pole.
Short answer: if you can see a clear spot in the sky between DEC +/- 70º and DEC +/- 90º (+ for northern hemisphere, - for southern hemisphere), and in addition, 60º in RA west of that clear spot is another clear spot in the sky, you can use ASIAIR's polar alignment process. If you do not mind reduced accuracy, you can extend the area in the sky to DEC +/- 60º.
Long answer: I have tested ASIAIR's PA on my mount simulator, and have found that the hour angle of the mount at the start and finish of the ASIAIR PA 60º slew does not affect polar alignment accuracy.
For example, place your mount's declination at somewhere between 89 and 90 degrees on the prime meridian side of the hemisphere (i.e., tilt your mount's DEC just a tiny bit towards the zenith).
Many people start at zero hour angle (i.e., OTA pointed almost at NCP), but that is not necessary.
In my tests, I have found that the starting hour angle does not matter at all. You can for example start with the RA that is 2 hours east of the meridian, and let ASIAIR slew it to 2 hours west of the meridian to perform the PA adjustments.
(In fact, the +2 HA to -2 HA configuration is most convenient for my RainbowAstro RST-135 mount because if I start at zero hour angle, it is difficult to lock one of the Altitude locking bolts when the mount has finished slewing.)
The ASIAIR only needs to plate solve before the slew, and again after the slew has finished (and while you are adjusting the altitude and azimuth bolts). So, as long as your OTA has a clear view of the sky at the start of the slew, and at the end of the slew, you are all set.
ASIAIR does not need images of the sky in between the starting and ending position of the 60 degree slew. And more importantly, ASIAIR does not need to see the pole itself.
From my simulation, I have not seen a polar alignment error of more than a few seconds of arc by starting at different hour angles.
For polar alignment, ASIAIR will also accept DEC angles that is as much as 30 degrees from the pole. It will tell you that you are more than 30 degrees off if you deviate too much. So now you have a huge piece of the sky to perform polar alignment -- 30º worth of DEC and 120º worth of hour angle.
However, try to stay above 80 degree declination if possible (+80 for northern hemisphere, -80º for southern hemisphere), because the accuracy of ASIAIR polar alignment appears to suffer when you get very far away from the pole. If memory serves, the error is large when you are 30 degrees in declination away from the pole (but even that might be acceptable, depending on your plate scale).
That being said, you do not need to have very precise polar alignment to capture good images. This is a simple calculator for field rotation (i.e., what happens if your RA axis does not point exactly at where the earth's pole points to):
http://celestialwonders.com/tools/rotationMaxErrorCalc.html
The result is in micrometers (microns). You can check your camera specs to find your camera's pixel size (typically between about 2.5 microns to 5 microns). If your error is less than the pixel size (or even the Airy disk of your optics), you do not need more polar alignment accuracy.
https://en.wikipedia.org/wiki/Airy_disk
The polar alignment accuracy will also depend on atmospheric refraction. This is not a problem if the polar region is more than 60º in altitude (i.e., your latitude is over 60 degrees). However for lower latitudes, atmospheric refraction compensation is less accurate, and you also need to factor in temperature (Bennett's formula uses 10ºC to estimate the refraction before temperature correction, for example) and pressure. I think the ASIAIR just uses "typical" temperature and pressure (I have never seen their code, so I don't know exactly what they use to compensate for atmospheric refraction.)
See here to learn about atmospheric refraction:
https://en.wikipedia.org/wiki/Atmospheric_refraction
You can see that it is completely moronic and a waste of time to try to adjust polar alignment for errors of less than 10 arcsecond, because the estimation error for the pole's location is greater than that, and with typical plate scales, field rotation will not be large enough to be noticed. Just because ASIAIR tells you the polar alignment error is zero, does not mean that your RA axis is actually pointed at the pole.
Good luck,
Chen