Frankly, I don't think Richard would use the name Alice/Alix/Alys for a daughter at all. Remember, it was also the name of his French fiancée who his father stole from him. Bad memories. If Richard and Berengaria did somehow have a third daughter, Sanchia/Sancha might be used, as I doubt Richard would care enough to dictate a name for her and Berengaria's parents were named Sancho and Sancha.
Yeah; this.
And as for Navarrese succession, IOTL Sancho VI's younger daughter's son succeeded Sancho VII, so I think Aénor wouldn't have much local problems there. When is she born again?
Arthur I of Brittany (b. 1186) m. Matilda I of England (b. 1191?)
A) Arthur I of England (b. 1209) m. Eleanor of Portugal (b.1211)
B) Richard I of Navarre (b. 1210) m. Margaret de Quincy, Countess of Lincoln.
C) Matilda of England (b. 1213) m. Louis IX of France (b. 1214)
D) Geoffrey I of Jerusalem (b. 1216) m. Isabella II of Jerusalem (b. 1212)
E) Constance of England (b. 1218) m. John I of Hainaut (b. 1218)
F) Berengaria of England (b. 1220) m. Rudolph I, Holy Roman Emperor (b. 1218)
I have to believe any first daughter of Richard would be named Eleanor; just like his pops, Richard was a total momma's boy.
I know the church gave a lot of grief about John's marriage to his cousin Isabella of Gloucester, so I'm not sure if they would be super keen on Eleanor (or Aenor Matilda) marrying her first cousin. It's clearly the marriage that makes the most sense politically however, it officially binds Brittany permanently to the Plantagenet inheritance.
In OTL, Richard did name John as his heir on his deathbed, despite having had Arthur as his heir prior to his own premature death in 1199. Perhaps his attitude changes with his own heir, though I doubt Richard would allow John to be regent or protector for her. Richard can't even really afford to grant John any further lands or titles beyond what he got from his marriage to Isabella, lest he grant John the means to raise money to support his own claim to the throne.
With Eleanor still the official ruler of Aquitaine, she could certainly name her granddaughter as heir to her own lands.
Arthur tho was Richard's heir presumptive, Aénor OTOH would be heiress apparent till the birth of a brother. I don't see Richard doing anything to bolster John's case ITTL