Partial restore to a different database
Is it possible to do this same partial restore, but to a different
  database?


Short answer I'm afraid, no.

Best bet is to do a partial restore to a new database then copy the tables and data to your target with:


SSIS Transfer SQL Objects Task. If this is a one off just use the import/export wizard in SSMS.
Use an INSERT SELECT script
If the tables are large, consider BULK INSERT.