Step 1: Evaluating statement I.
Second ionization enthalpy of Be is actually higher than Li because removing an electron from fully filled 2s orbital in Be is more difficult. Hence statement I is false.
Step 2: Evaluating statement II.
Hydration enthalpy increases with higher charge density. Be\(^{2+}\) has smaller ionic radius and higher charge density than Mg\(^{2+}\), so Be\(^{2+}\) has greater hydration enthalpy. Hence statement II is false as given.
Step 3: Evaluating statement III.
Li\(^+\) has more electrons and larger size compared to Be\(^{2+}\). Thus Li\(^+\) has greater ionic radius than Be\(^{2+}\). So statement III is correct.
Step 4: Final verification.
Only statement III is correct based on periodic and ionic size trends.
Step 5: Final conclusion.
Correct option is II, III only
Final Answer:
\[
\boxed{\text{II, III only}}
\]