If you raise the power and your SWR reading changes one of a few things is happening.
1) You either have a problem with the antenna itself, or the added power is pushing one or more components near or beyond their limits, or the antenna is in the process of failing on you.
2) The amplifier being used is not putting out a clean signal, and thus is spewing out harmonics and/or spurious emissions. This could be an issue with poor amplifier design, or could be the just as common issue of the amplifier being pushed to hard in any number of ways (to much power driving it, running at a voltage beyond the design specification
(1), ect.) An antenna system that is wide banded will have less of a change in the SWR readings in this case as it doesn't try as hard to reject out of band signals as a narrower banded antenna.
A note on the issue being with a magnet mount. A magnet mount is essentially a capacitor between the magnet and the metal of the vehicle body underneath. Increasing power does not affect the capacitors ability to pass AC current, to a point. This point is when there is enough voltage to arc across the capacitor plates. As the same setup and the same mount are not having this issue with another antenna it is definitely not the issue here.
The DB
(1) When I say design specifications here I am not referring to the amplifier but the transistors inside. It is common practice for companies such as X-Force and Dave Made to recommend you far exceed these specification. There is always a cost to exceeding the specifications on said transistors, even if you don't notice it.