Perhaps the simplest solution would be to extend your criteria a bit.  If the day of the month is less than or equal to 5, include LastFullMonth and MonthToDate.  If not, just MonthToDate.  But that is moving the goalposts.

The general issue is identifying that January 1 is not a business day.  Identifying weekends is simple.  Identifying holidays is not.  After all, many businesses have different holidays from the standard federal/banking holidays (and that's not even taking international considerations into account).

Most solutions involve creating a Holidays table in your database, and doing a sort of negative join with this table.  Do a standard left join from your primary data source to the Holidays table, but only select records where the Holidays date is NULL.