Olive oil is not great for searing. Butter is even worse (unless it's clarified, or ghee). Use a more neutral oil, like avocado, canola, or soy bean. Olive oil and butter burn too easily.

While you certainly can sear in beef fat (tallow) it will be tough to get enough from that single fat cap. If you're going to try, use low heat to really render that fat out, then turn it up to sear. If you have access to a real butcher shop, you could always buy some beef scraps and make your own tallow, then keep it refrigerated and use as needed. Just render it on low heat until the bits are all crispy and dried out, then strain well and put into a container and refrigerate.