Can you make Bearnaise with olive oil?
Don't make any emulsion sauces with olive oil. The high speed of dispersion, especially with an electric blender, creates unpleasant bitter compounds in the oil. But frequently, even the speed of a hand whisk is high enough to ruin the taste. Either that, or you are too slow to create the emulsion at all. 

You can form yolk-fat-emulsion sauces with practically any fat, and olive oil is an unfortunate exception with its taste change. Just choose another fat compatible with your diet. From a purist's point of view, they won't be a Bearnaise, the French sauces are very tightly classified. You will also get an obvious difference in taste (only butter tastes like butter), and, if you are using a liquid oil, a slight difference in handling - you can let a liquid-oil-emulsion cool down, while a butter or lard based sauce will harden into an unpleasant grainy mass. 

If your base food is very plain/bland tasting, you can play with subtle oil tastes to create an interesting effect where the main "seasoning" is the oil in the sauce.