Add missing column to SSIS, data not extracted to staging table
In addition to adding the new column to the variable User::vSQLBase, you need to open the Data Flow for the staging table being loaded. Edit the source, and click on Columns, verify you see the new column, click OK. Then edit the Destination, and click on Mappings - and make sure that the new column from the source is pointing to the right column in the staging table. That will do the new column mapping the consultant was talking about. You'll need to make sure the mapping is updated for whatever moves data from staging to the destination table too. Then save and deploy again.

An incremental load only populates new or changed data since the last successful run. Do you care about the new column being populated in the destination only going forward? Or is it important for that data to exist in records previously imported as well? For an accounting system, I'd guess you'd want the new column populated for all the records in the table. In that case, I agree with your consultant that a full load is necessary.