Yeah, with complex survey designs everything is harder, so most the stats tests in SAS are very hacky. You are right in running a regression, and it should give you the right results. The DOMAIN specification makes sense here as well.
I found this link by searching "PROC SURVEY T-TEST": https://www.cdc.gov/nchs/tutorials/dietary/Advanced/EstimatePrevalence/Task4b.htm
If you plan on doing this more often, you might want to try this coursera class: https://www.coursera.org/learn/data-collection-analytics-project