It is impossible to maximize rivalry with Bethany.
Rivalry with Bethany can generally be earned by choosing aggressive responses, by siding against mages or by exploiting Fereldans.
After defeating Corypheus and returning to your house, talking with Bethany and choosing the diplomatic or sarcastic options yields ( +5). Legacy (quest) If you side with Larius ( +5). Wayward Son: If Hawke choose the option "make up for your neglect" when speaking with Vincento. Tranquility: When asked to kill Karl, choosing "Do as he asks" yields ( +5) Tranquility: When talking to Anders, choosing "You want me to fight templars?" yields ( +5) Alternatively, deferring to Bethany will result in the same approval gain, though Aveline Vallen does not gain ( +5) also. Tranquility: Upon leaving Lirene's Fereldan Imports you will be approached by Angry Fereldan choosing the response, "We're on the same side." (available to a Diplomatic Hawke) instead of deferring to Bethany results in ( +5). Tranquility: Promise Lirene you won't harm Anders ( +5). Shepherding Wolves: Refuse to hand over Ketojan to the Arvaarad ( +10). Shepherding Wolves: Accept the quest from Sister Petrice ( +5). Portrait of the Past: Giving Bethany your mother's portrait (found in a chest in the Amell Estate basement) back at Gamlen's House results in ( +10), regardless of dialogue choices. Loose Ends: (Athenril/Smuggler Path) Let the young boy start a new life with his sisters by letting him have the loot. Long Way Home: When Merrill opens the barrier with blood magic, choose the Aggressive option, "The demon, you mean." for ( +5). Enemies Among Us: Ask for Bethany's assistance in breaking Idunna's compulsion ( +5). The Deep Roads Expedition: Take Bethany on the expedition ( +15). A further ( +10) can be gained upon returning to Gamlen's House with Grandfather's Will and speaking with Bethany by choosing the friendly response on the second dialogue wheel.
Birthright: By simply entering the vault in the Amell Estate basement you gain ( +15). Bait and Switch: Directly defend Bethany at the end of the quest for ( +5). The Destruction of Lothering: As a non-mage, choose the Aggressive response to Wesley's threatening of Bethany, upon meeting Aveline for the first time for ( +15). Bethany is a faithful Andrastian and also distinctly opposed to blood magic, so rebuffing demons and opposing blood mages will earn friendship as well. Friendship with Bethany can generally be earned through acts of kindness, by siding with mages or by supporting family.