One is that I don't think it follows from the premise that the continuity of the physical world precludes AI, brain scanning, etc. Even if the physical world were continuous (likely not, see below), an arbitrary degree of approximation could be attained, in principle. At the very least I would not call the footing "firm".
The second is that the universe is very likely not continuous anyway. The Beckenstein bound[1] puts an upper limit on the number of bits of information a region of space may contain. If the ruler tickmark were either measured or localized to the precision required to encode the information, the information density would cause it to collapse into a black hole. This would happen once your measurement needs to be about as precise as a Planck length, which would allow you to encode about 115 bits of information with your tickmark.
(This in of itself is independent of the fact that you would need to construct the ruler out of objects that the universe permits; your ruler tickmark would need to be made of and measured with discrete fundamental particles, which by their very nature are quantized).