Phase-sensitive measurements of direct-current superconducting quantum interference devices (dc-SQUIDs) composed of Sr2RuO4-Ru eutectic crystals have been performed to temperatures below a bulk Ru superconducting transition temperature at 0.49 K. A SQUID with Nb/Ru/Sr2RuO4 junctions fabricated on one Ru inclusion exhibits two distinct transitions due to the Ru superconducting transition and competition of proximity-induced superconducting gaps at the junctions. At sufficiently low temperatures, the SQUID interference patterns start to collapse with large phase shifts of the Fraunhofer patterns. This result indicates the influence of magnetic fluxes induced by large bias currents flowing in a strongly asymmetric supercurrent path. Such a large change in supercurrent path suggests superconducting phase mismatch between the s-wave and chiral p-wave states at the Ru/Sr2RuO4 interface.