The Beast Rider cavalier may take a mount that functions as an animal companion. The companion mount does not gain the Share Spells feature, and instead gains the "Exotic Mount" ability:
At 1st level, a beast rider forms a bond with a strong, loyal companion that permits him to ride it as a mount. This mount functions as a druid’s animal companion, using the beast rider’s level as his effective druid level… A beast rider’s mount does not gain the share spells special ability.
(ellipsis added by me)
There is also the Charger archetype for animal companions. A companion with this archetype has the "Mounted Challenge" ability:
When ridden by a cavalier, chargers gain half of the bonuses and penalties granted by the cavalier’s challenge class feature.
This ability replaces share spells*.
Since this companion archetype was intended for cavaliers, the asterisk refers to the following note:
A cavalier’s mount should therefore be treated as having the share spells feature for the purpose of qualifying for animal archetypes.
But an exotic mount explicitly loses the Share Spells feature; if the Charger archetype cannot replace it, then that suggests incompatibility.
On the other hand, maybe there is no conflict by the class archetype rules. Beast Rider and Charger apply to separate creatures, and the mount is not really a class. Also, Beast Rider is one of the few cavalier archetypes that has a companion mount, whereas regular mounts would not be applicable for an animal companion archetype in the first place; it seems like these archetypes were intended to be compatible.
Note: Pathfinder Society rules do not apply in this scenario.
Are theses archetypes compatible, or is there a conflict?
Best Answer
From a strict reading, the mount does not have Share Spells (even though it's an animal companion, and they usually do) because it's a Beast Rider mount (different from normal Animal Companions). Note here that it does not say that it loses the Share Spells ability, merely that it does not gain it. However, it is also a Cavalier mount, and therefore should be treated as having Share Spells for the purpose of qualifying for archetypes. (This is a specific exception, as Cavalier mounts usually do not have that ability. This mount also does not have that ability.) There does not appear to be a conflict here.
By extension, RAW it should be able to qualify for animal archetypes (and only animal archetypes) as if it had Share Spells.
Technically, if you really wanted to cheese it for RAW, you could argue that it would continue to qualify as such even if you had taken another archetype that also removed Share Spells, but that, I think, would be pushing things.