Check if you are minimal logging

A LOP_INSERT_ROWS will be present for every inserted record if you are not minimal logging

SELECT Operation,AllocUnitName FROM fn_dblog (DEFAULT, DEFAULT) WHERE Operation = ‘LOP_INSERT_ROWS’

Running SSIS package under different credentials

Create windows user name which will run your package
Create a sql login for the windows user giving it the relevant permissions
Create a credential for this login
Create a proxy that uses this credential
Update your job step to use the proxy