I will try to address your statements in turn as I don't think I am doing a good job of articulating my thoughts. That said, I think we are closer than I originally believed.

To me that whole section is justification for why to reserve the right to suppression: because the intolerant may not engage with argument in good faith, and they may instead tell their followers to denounce logical argument altogether and be violent.

The section justifies when it may be reasonable to suppress incitement to violence. The "and" operator requires the incitement to violence in the premise: "But we should claim the right to suppress them if necessary even by force; for it may easily turn out that they are not prepared to meet us on the level of rational argument, but begin by denouncing all argument they may forbid their followers to listen to rational argument, because it is deceptive, and teach them to answer arguments by the use of their fists or pistols."

Popper did not write "or." Both incitement to violence and rejection of rational argument is the [non-exhaustive] example Popper uses to justify the right of suppression.

The key word here is may, which to me means “this might happen”. I basically read this as a warning of a potential outcome of intolerant philosophies - and why suppression might be needed and why to reserve the right to it - but not a threshold for when to suppress them.

I agree. As above, the example he used it not explicitly exhaustive. There may be other reasons to claim the right of suppression. However, Popper never used any other examples as valid cause to suppress speech, and in the immediately prior statement, cautioned that suppressing democratic speech is "most unwise."

Given this, the implied claim that Popper and/or the Paradox of Tolerance supports the categorical suppression of "intolerant" speech (as the comic does) are inaccurate.