I would b careful with the TS-590's SWR announcing capability.
I would suspect that it would only apply if you had an antenna directly connected to 1 of its SO-239 antenna jacks.
Using an external antenna tuner, you would only be getting the SWR between the rig and tuner, which should be no more than 1.5/1 or flat and if not, I would think there might be something wrong. Between my rig and amp, my TW-1 reports an SWR of 1.3/1 and 1.0/1 between the rig and tuner.
The 570 reports 1.5/1 with both units. To get an accurate measure, I'd use an external talking meter between the ATU and antenna and just forget about the 590'S SWR announcing unless an antenna was directly connected.
73:
Mike VO1AX