Assign Custom Record Types in Permission Sets

You can assign record types to users in permission sets.
Available in: Salesforce Classic (not available in all orgs) and Lightning Experience
Record types available in: Professional, Enterprise, Performance, Unlimited, and Developer Editions

User Permissions Needed
To assign record types in permission sets: Manage Profiles and Permission Sets
  1. From Setup, in the Quick Find box, enter Permission Sets, and then select Permission Sets.
  2. Select a permission set, or create one.
  3. On the permission set overview page, click Object Settings, then click the object you want.
  4. Click Edit.
  5. Select the record types you want to assign to this permission set.
  6. Click Save.