I would like to know if we can determine the fracture toughness properly by performing a fatigue test (high cyclic loading on a Compact Tension specimen, with crack growing?)
You probably know that the fatigue properties of a material are measured during a test in which the load is cycled and the fracture toughness of a material is measured during a test when the load is monotonically increasing. Therefore it is not generally possible to measure toughness during a fatigue test.
You also may know that a toughness test is generally performed on a specimen which has a "hairline" pre-fatigue crack. That is, the crack tip has been generated by fatigue so that it is sharp. A sharp crack tip is likely to give lower/conservative levels of toughness (e.g. see figure 1 here: https://www.twi-global.com/technical-knowledge/published-papers/the-effect-of-notch-sharpness-on-the-fracture-toughness-determined-from-sent-specimens).