Why is the Beluga chartered out for oddball missions, but not the Dreamlifter?
The 747LCF cannot haul any cargo except Boeing’s own cargo as its type certificate only allows it to carry Boeing’s own cargo.

Whereas A300 Beluga’s type certificate limitation section only refers to Airplane Flight Manual. It may allows to carry other cargo with some conditions met.

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Page 28 of A20WE, type certificate of 747, states


  XIV - 747-400 Large Cargo Freighter (LCF) Major Design Change (Approved June, 2, 2007 ) Transport Category
  
  Allowable cargo
  
  These airplanes are not approved for commercial freight hauling operations of material other than that approved per Exemptions 8769, 8769A and 8769B. Only cargo that supports Boeing corporate lines of business is allowed for carriage. All items intended for carriage must conform to the standards found in Document D926U013-44, “747-400 LCF Flammability Acceptance Criteria for Cargo Carriage,” or be accepted by the FAA once a safe method of transport has been established. A summary of all items allowed for carriage is identified in Document D451U742-01, “Allowable Cargo – 747-400 Large Cargo Freighter.” Document D451U742-01 is considered part of the Weight and Balance Manual/Airplane Flight Manual. In addition, a listing of the FAA-approved shipping mechanical equipment (SME) fixtures that are approved for installation on the 747-400 LCF aircraft are contained in the Weight and Balance Control and Loading Manual (Document D043U545-BHC1).