Precise measurements of the branching ratios for the flavor-changing neutral current decays $K\to\pi\nu\bar{\nu}$ can provide unique constraints on CKM unitarity and, potentially, evidence for new physics. It is important to measure both decay modes, $K^+\to\pi^+\nu\bar{\nu}$ and $K_L\to\pi^0\nu\bar{\nu}$, since different new physics models affect the rates for each channel differently. The goal of the NA62 experiment at the CERN SPS is to measure the BR for the charged channel to within 10%. For the neutral channel, the BR has never been measured. KOTO, an experiment at J-PARC, should have enough data for the first observation of the decay by about 2021. We are designing the KLEVER experiment to measure BR($K_L\to\pi^0\nu\bar{\nu}$) to $\sim$20% using a high-energy neutral beam at the CERN SPS starting in Run 4. The boost from the high-energy beam facilitates the rejection of background channels such as $K_L\to\pi^0\pi^0$ by detection of the additional photons in the final state. On the other hand, the layout poses particular challenges for the design of the small-angle vetoes, which must reject photons from $K_L$ decays escaping through the beam pipe amidst an intense background from soft photons and neutrons in the beam. Background from $\Lambda \to n\pi^0$ decays in the beam must also be kept under control. We present findings from our design studies, with an emphasis on the challenges faced and the potential sensitivity for the measurement of BR($K_L\to\pi^0\nu\bar{\nu}$).