You were wrong. The CMRI and Bishop Dolan/Fr. Cekada do have some disagreements on a few matters, but they do not oppose each other nor is there, as far as I know, a "rivalry". Bishop Pivarunas as spoken well of both men at the CMRI's Fatima Conferences. It is true, however, that the two groups don't generally talk about the other.
This is my understanding.