The paramount purpose of the Roman census was for raising taxes. (There was also the secondary purpose of identifying and locating able bodied young men who could be impressed into armies and labor forces for large construction projects as needed.) There were two types of taxes levied in those times: Hearth taxes, similar to our property taxes, levied on a household which might be calculated based on the number men, women, and children occupying the household: and a "business" tax that was usually paid in kind as a percentage of ones crops or shop production, or in the case of a carpenter like Joseph, a certain amount of labor contributed to the state, but that also could be paid in coin it the taxee could afford to do so.
The point is that for the purpose of taxation (and also of locating able bodied personnel), the census had to count people at their places of residence and work! The taxing authorities wound not care about the taxee's ancestry or place of birth and certainly would not want or allow people to leave their current home towns until the census was completed. This idea of Joseph and Mary being required to travel to their (or his) birthplace for a census is totally bogus and nonsense. Even if the taxing authority, for some ridiculous reason, wanted to tally people by their place of origin, there would be no need to make them journey to that place; they would just ask them where the hell they were born. The significance of this information to the census would be miniscule anyway since only a tiny fraction of the population in those days ever spent any time away form there place of birth.
The census story only serves to combine some individual born in Bethlehem with another individual who came from Galilee, i. e., it lends credence to the theory that the Jesus of the new testament is a composite mythical invention based on the lives of two or several self-proclaimed prophets/messiahs of that time. Remember that prophesy and salvation were the MTV of those times and the streets were full of their practitioners (I wonder if they had a union and demanded coffee breaks).
How about the fact that woman slipping and falling 2 feet (distance from hips to ground) can cause fetal demise? That's not even taking into consideration the amount of jostling she would experience on an 80 mile journey. This would take 2 weeks of horseback travel.
Even if fetal demise did not occur, it is almost certain horseback riding would certainly induce labor*. *Riding horses during pregnancy
If you really want/need references stating that horseback riding is not recommended while pregnant
But you are ignoring the fact that the fetal Jesus was of divine origin and would be immune to such problems.