There is no such thing has a "part 91" type rating. It does not exist.

There is no such thing as a "part 135 type rating."

You have a pilot certificate. On that certificate, you hold a type rating. Whether you conducted flying with that type rating under Part 91 or part 135 previously, is irrelevant.

An operator must adhere to the requirements of it's FAA-approved training program. If the operator uses a contract training center and has no provision for the short course, then you may be required to undergo the full training. Many operators have provision in their training program to account for existing certification. Many do not, or require the full training program.

Operators often prefer or require previously typed applicants, as it costs the operator less for training expenses.

You'll be required to undergo whatever the operator requires per its approved training program.