I don't think the #6 secondary wire would be fine if you increase the secondary breaker size. It might not be large enough even using the 60A breaker size. The specs are listed as 40A, and use #6 wire. 40Ax1.25=50A. #8 has an ampacity of 50A at 75 deg, so lets assume they are using the 60 deg column, and the #6 has an ampacity of 55A.
Per 240.21(C)(1), the secondary conductors can be protected by the primary OCPD on a delta-delta, 3 wire transformer if the primary OCPD value does not exceed the secondary ampacity x sec/pri xfmr ratio. 55A x 0.5 = 27.5. The 30A (or 40A or 50A) primary breaker would not work for the 55A ampacity of the #6 secondary conductors.
If instead, you wanted to use the secondary breaker to protect the secondary conductors per 240.21(C)(2), then the ampacity of the secondary conductors must not be less than the rating of the OCPD where the secondary conductors terminate. The 55Amps of the #6 is less than the 60A c/b.