It sounds to me like it could be a "common mode current" problem, RF on the outside of the co-ax - from an unbalanced load, or maybe some other reason. This 'common mode' current can mess up your SWR reading, but the 'actual' SWR might still be OK, but the measurement system is being interfered with.
If your coax is long enough, try coiling up 6 or 10 turns of the coax, 3 or 4 inches in diameter, and see if it makes a difference.? Or put snap-on ferrite beads on the coax (near the antenna).? Or if you have a large enough toroid, like an FT240 or any size large enough to pass your coax with connector through a few times, use that.? I have my station coax wrapped 8 turns through an FT240-43, and the PL259 still fits through.? If you're using a BNC cable, and FT140 size should work.
If you've got spare time, read up on 'common mode choke' for more and better info.
I have a common mode problem (I think) on 6 meters (same antenna and feedline as all HF bands).? Keying up FT8 on 6 meters switches my rig back to 'Receive' and often disrupts the USB connection to the radio.? Other bands on the same equipment are fine.
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Doug, K8RFT