Why might my psychiatrist refuse to prescribe me medications for my chronic pain?

Depends: It depends on how skilled your psychiatrist is at prescribing pain medications and monitoring for its effectiveness.

Dr Williams agreed

Pain medications: Many psychiatrists are less familiar with pain medications and a good number of them can be dangerous in overdose and habit-forming if not used carefully. Additionally, some pain medications like Suboxone require special licenses. Your psychiatrist may want you to seek someone with specific expertise in pain management and medications.

Psychiatry: While it is true that your psychiatrist may not be comfortable or familiar with all the pain medications and he may want you to see a pain specialist for this - it is also true that he may also feels that based on examination pain medications are not justified or based on the psychiatric diagnosis your request for pain medications may be related to this diagnosis - talk to him!