The Raffles Hotel in Santa Monica or
Ocean and Vine at the Loews Santa Monica Hotel have a very nice setting, on the beach, with a very nice breakfast at a fairly high price (Ocean and Vine's breakfast is well done, but nothing inspired as far as the choices). Not directly between LAX and the Getty, but not too far out of your way.
A better breakfast for a better price can be had at
Jinky's in Santa Monica. But not on the beach.
Or you could grab a donut and coffee at
Stan's Donuts in Westwood, and munch on it while wandering Westwood Village. My favorites are the Reese's Peanut Butter Pocket (a donut stuffed with peanut butter and coated in chocolate and chocolate chips), or the Cherry Cream Cheese Donut (stuffed with a sweetened cream cheese filling and a danish-style pocket of cherry pie filling on top). Also not on the beach.
There are also some nice breakfast restaurants on the Third St. Promenade in Santa Monica. Wander and see what catches your nose/fancy. Not on the beach, but with a two block walk you will be on the beach and can get to the pier/boardwalk with another two block walk along the beach.
The Daily Grill in Brentwood also has a decent breakfast menu and great chicken hash. That's on your way, but not on the beach.
If you want to skip the Museum Cafe lunch, stop by
Versailles in Culver City (at Venice and Motor) for some of the best Cuban food in LA (Get the garlic chicken or the lechon asado, plenty of Plaintains and maybe a black bean and rice combination, cooked together, as your side dishes) at really reasonable prices. There can be a line after 4 pm, but if you get there at about 1 or 2, you should get seated right away, they serve quickly, and you can be back to the airport within about 20 minutes of leaving the restaurant.