It's not even clear that we can do experiments that will definitively tell us what the early universe was like. Inflation answers a lot of questions, and gives testable predictions, which are confirmed. If inflation also implies that there is a multiverse and not a universe, it is an implication of the theory. To argue against it definitively you'd have to come up with a new theory. What type of anomalies are you referring to? Surely not the axial vector anomaly? If one is led to the existence of a multiverse by a string of consistency checks, we have no choice but to accept it, unless another theory comes along which better explains the data.