How can I get a row count in DBI without running two separate calls to process?

Based on a quick look here, it seems that after you run

$statement->execute($arg)

you can access the row count via

$statement->rows

The "caveat" in the documentation (linked to in a comment on another answer) is important, and provides the real, correct answer:

Generally, you can only rely on a row count after a non-SELECT execute (for some specific operations like UPDATE and DELETE), or after fetching all the rows of a SELECT statement.

For SELECT statements, it is generally not possible to know how many rows will be returned except by fetching them all. Some drivers will return the number of rows the application has fetched so far, but others may return -1 until all rows have been fetched. So use of the rows method or $DBI::rows with SELECT statements is not recommended.