As stated, the Red Heifer sacrificial process involved taking the animal to the Mt. of Olives opposite the temple.
Was a reason for this ever given? Is it purely to emphasize the separation of this specific sacrifice from the others or was another reason given for why they went out of their way to bring it there?
The reason it was done outside comes from the verses:
And you shall give her to El῾azar the priest, that he may bring her outside the camp, and she shall be slaughtered before his face:
And El῾azar the priest shall take of her blood with his finger, and sprinkle of her blood towards the front of the Tent of Meeting seven times:
Rashi there explains that outside the camp meant outside the three camps (Tabernacle, Levites, and Israel). In the Temple times, that translated to outside the city of Jerusalem. And the sprinkling of the blood had to be with a view of the door to the Temple sanctuary. As that was on a mountain, it would take being on another mountain to see it.
It is more likely the Christians deliberately made the story of Jesus occur on Mt. Olives in order to make the connection seem more apparent, as they did with the rest of the story. They decided to make Jesus connected to the red heifer, so they wrote the story with him killed on Mt. Olives. They act like the archer in the famous parable of the Dubno Maggid:
A man found an archer at an archery range who had only perfectly accurate shots. When the man asked the archer how he had such consistent accuracy, the archer responded that first he shot the arrow and then he painted the target.
As a technicality, the reason could simply be because the Para Adumah (red heifer) was not a korbon, and one may not slaughter any non korbonos inside the Beis Hamikdash.
Another problem with doing it in the Beis Hamakdush would be that all involved in the process become impure, and one may not be in the Beis Hamikdash in impure state.
The Ramban Bamidbor 19:2 says the fact it was done outside is what causes the non-Jews to question it and compare it to idol worship. However the reason it was done outside is that bringing it outside purifies the outside world from impurities. (don't know what that means-The Ramban in his introduction to Chumash says that those who don't understand Kabala won't understand the kabbalistic references in his work on Chumuash even if they understand the translation of the words)
