First verify that the power/SWR meter calibration is correct for
both the meters (in the radio and in the tuner). There are usually
two steps: first, a trimmer capacitor or similar at the SWR detector
that is adjusted for minimum reflected power into a good 50 ohm
dummy load. Second, the power sensitivity settings for each
power range (forward and reverse) need to be calibrated against
a known standard.
If you get the same readings on both when the radio is connected
directly to the tuner (rather than through the amp), then that likely
isn't your problem. But you may need to use different power
Tuning up at too low of a power can also cause such a problem,
due to the forward voltage drop across the diode detector. I see this
sometimes with QRP rigs, where the reflected power level is too low to
get a good reflected indication if the output is reduced, and the SWR
appears to increase with power because the reflected power reading
becomes more accurate. This might be the case if you are using the
power meter in the tuner on the high power range (as would be the
case when using the amp), as it might not register reflected power
properly at lower levels, so the SWR would look better than it
actually is. But the meter in the radio should still give an accurate
indication.
Otherwise a likely cause is that the path through the amp in bypass
mode isn't a perfect 50 ohms impedance. That may vary with
frequency, and some amps will be better than others.
The same could if the extra jumper cable you use to go through the
amp isn't 50 ohms: I had a jumper cable of RG-59 (75 ohms)
that I got confused with RG-8X (they are the same diameter) that
changed the SWR. Again, the difference will be highest on the higher
bands.
When the amp is actually in use, the SWR seen by the rig is that
of the input to the amplifier, and should only be slightly affected
by the settings on the tuner (if at all).
Edited because I just noticed this part:
Linking in the amp and the tuner - even when I bypass them - leads to a slightly higher SWR as seen near the radio.
So the antenna has a low SWR when connected directly to the
radio? Does that change with the power level?
If so, do you actually need the tuner at all?
That also means that there probably are at least 2 different
patch cables added into the mix if you are adding both the
tuner and the amp at the same time.
A small change in SWR going through an amp or tuner in
bypass mode isn't unexpected - they often aren't designed
for a clean 50 ohm path (especially on higher frequencies).
When in doubt, adjust the tuner for lowest SWR at the radio
if you are operating with the amp bypassed. That should
account for any variation in the RF path through the
equipment.
When using the amp, adjust the tuner for minimum SWR as
seen at the amp. The SWR seen by the rig is determined
by the input circuitry of the amp in that case, not by the
setting of the antenna tuner.