Why Commonwealth membership does not control Solomon Islands policy

Can Britain prevent Solomon Islands from making security arrangements with China because it belongs to the Commonwealth and shares the monarch?

No; Solomon Islands controls its own foreign policy, unlike a British overseas territory whose security Britain guarantees.

A Commonwealth country is distinct from a British overseas territory. Solomon Islands may retain the British monarch as head of state, but Britain does not control its foreign policy. An overseas territory would trade that control for a British guarantee of security and defence; the independent state has made no such exchange. Its government can therefore pursue a Chinese security agreement, even if its attempt to play competing powers against each other appears ineffective.