Short answer
HubSpot's KB: a single-line text property maps only the Record Type ID. A dropdown property (advanced) can show the label. After mapping, resync contacts to fill the property; new record types must be added to the dropdown manually.
1. In HubSpot
The relevant screen in my demo portal.
2. Worth knowing
Pick dropdown if people will filter by record type.
3. Related
Sync rules and integration settings.