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According to the web site of the South Carolina Department of Natural Resources, either a SSN or an alien identification card is required:
Social Security Number: The disclosure of your social security number or alien identification number is required to obtain SC hunting and fishing licenses. This complies with SC Code of Laws Section 63-14-1080 and Federal law 42 USCA 666(a) (13), which requires a licensing agency to provide this information to the Child Support Enforcement Unit of the Department of Social Services to establish, modify and enforce child support orders.
There is a non-resident license application for mail-in applications, but it requests a US address.
It is unclear to me whether non-resident licenses are available only to US residents (who live outside of SC), or to anyone, including US non-residents, but it appears (based on the above quote about US Federal Law, and the application form) that a licence likely is not available to non-US residents.
To be sure, you may need to contact the office directly.
Another option might be to find a private lake/pond where you can fish without a license.