Step 1: Recall the stability trend.
For group 2 carbonates, thermal stability rises as we go down the group. This is because the metal ion gets bigger and pulls less on the carbonate.
Step 2: Link size to polarising power.
A small cation has high polarising power. It distorts the carbonate ion strongly and helps it break into an oxide and carbon dioxide.
Step 3: Find the smallest cation.
Among Mg, Ca, Ba and Be, the beryllium ion $Be^{2+}$ is the smallest. So $BeCO_3$ has the least thermal stability.
Step 4: See what this means.
Because $BeCO_3$ is so unstable, it tends to break down into BeO and $CO_2$ even at room temperature.
Step 5: Explain the $CO_2$ atmosphere.
By keeping it under a carbon dioxide atmosphere, the extra $CO_2$ pushes the reaction backward and stops the carbonate from decomposing.
Step 6: Pick the answer.
So the carbonate that must be stored under $CO_2$ is beryllium carbonate. \[ \boxed{BeCO_3} \]