Record Type in flow is actually stored, realized and worked through as a separate object, which is technically how it is kept in the Salesforce databases. It is not natively a picklist on the original object; Salesforce just presents it that way as a matter of convenience for the users.
Therefore, when working with Record Type in Flow, you would not be able to access it as a standard picklist, since it is not really one. There is, however, the object "Record Type" that can be set when performing any Record Lookup.
My advice would be to create a non-Picklist Choice field on your screen, just a normal Choice, and populate it manually with the possibilities. You can then use a Decision to set a path based on which Record Type was chosen, and in each possible path, run a Record Lookup on the Record Type object, getting the ID of the Record Type where the name matches the choice. Actually, that is the easy path, but a bit more fragile; it would be better to perform the lookup based off of the DeveloperName of the Record Type, though you will have to manually enter these into your Decision criteria.
Once you have the Record Type ID, you can use it in the Flow wherever you need it.