She's 100% Andoran and Elayne's stated policy is to avoid that impression that Andor has taken over Cairhien since she took the Sun Throne. Which most likely means she'd have Cairhienians as her top advisers in Cairhien and the possible regent figure would also be a local, though with Travelling available the latter might be unnecessary.

There's also the Andoran invasion of Cairhien back in TFOH to consider - it was Rahvin's doing, but it was done in Morgase's names, and surely made her quite unpopular there.