It looks like FIR Designer matches REW in mag and phase. The measurement clearly has very high phase slope and thus significant bulk delay. This bulk delay should be removed in the measurement process if phase is to be used in any subsequent filter design. FIR Designer does its best to create an IR from the freq/mag/phase TXT format but using freq/mag/phase looses information when the measurement has very high phase slope. Better to export the measurement IR directly and load this into FIR Designer.
BTW, from a display perspective, in Preferences you can uncheck “Show phase plot wrapping” to remove the vertical lines between when the phase wraps.