When it comes to the steak, I think something a little thicker is needed—1/2" minimum. What's important here isn't necessarily the cut of beef itself but, rather, the thickness and whether or not it has a bone (see recipe note for more on that).
When I make this, what I typically do is salt the beef and let it come to room temperature for about an hour on a wire rack set over a sheet tray. I put a cast iron pan in the oven, set it to 450° and leave it in there for 30 minutes once it comes to temp. I start the béarnaise, stopping just before melting the butter. Smear the steak in oil, heat the cast iron on the stove over medium-high for about 5 minutes and then slap the meat in the pan to sear it. This is when I start to melt the butter.
If the steak is 1" thick, I flip it over and immediately top it with a pad of butter and slide into the oven; if it's less than 1" I usually just cook the whole thing on the stove (see recipe notes below). Either way, during all this is when I emulsify the butter into the egg yolks, and stir in the tarragon and lemon. By the time the steak is finished resting, the sauce is done (and, if you throw a russet potato in the oven when the pan goes in and you keep everything in there a bit longer, you'll have a good starchy side, too),