[@Ever Faithful]: I'm going to reject this form out of hand for a variety of reasons, detailed below.

[list][*]Eustace is, on a fundamental level, a character who does what he is able to do simply because of the nature of the medium he is from. Courage the Cowardly Dog is a comedy at heart—one that centers around a dog that is able to overcome all odds against all manner of horrors and supernatural beings, granted, but a comedy nevertheless. To claim that a character would be able to perform superhuman actions based solely off of the fact that the medium that he is from (more specifically, a cartoon comedy show that plays with extreme violence for laughs and wild gags for comedy) is ignoring the context in which the character is presented. By the same logic, could Tom and Jerry (from the titular cartoon) not be as gods unto men (or, well, cats and mice in this case) despite the comedic violence they are famous for being little more than just that?
This is hyperbole, of course, but when everything else in the world (and all the characters therein, both from it and not) plays by similar rules, would it not be reasonable to assume that anything that so exceeds the threshold for one's suspension of disbelief that it wholly breaks immersion would not lend itself to breaking the semi-cohesive setup that is [i]already[/i] reliant upon some basis in feasibility to function? A character needs to be presented as serious enough so as to not simply devalue or ignore the cohesion of what has been presented and what every other person involved in the RP has "bought into", so to speak. On that note...
[*]The character does not mesh with the cast as far as I can see, nor does he have any reason to mesh with them [i]nor[/i] any reasonable willingness to follow along with their actions, whimsical or otherwise. Eustace is, as you have noted, a crotchety, grumpy old man who is prone to weakness and cowardice at his core. He's stubborn and seldom cares about anyone beyond himself, Miriam, and [i]maybe[/i] Courage at times. In what world would someone like that ever be willing to place themselves on the front lines and fight instead of just letting someone else resolve it for him, [i]especially[/i] when they show themselves to be more capable of handling what has been thrown at them?
This is ignoring the fact that Courage the Cowardly Dog, as a cartoon show aimed at children, does not present Eustace's actions as anything more than comedy that oftentimes result in his comeuppance. Nobody is ever meaningfully hurt, nobody ever dies; who is to say that a person with those traits mentioned above—and who might otherwise may as well be a "normal" human at heart—could ever survive in a world like this to begin with?
[*]Linking back to the first point: being from an episodic show with no intent on continuity beyond the occasional two/three-part stories (after which everything returns to status quo [i]anyways[/i]) does not, in my eyes, present itself as a valid argument for abilities in stories where the world is persistent and the actions of characters have meaningful, long-term consequences. Eustace is abused and ends up in ridiculous situations (often of his own making) because the setup of the show often desires it. It's not "fun" to see a character so mean to a protagonist and everyone around him escape consequences, so the show punishes him for his hubris. But with so many of those punishments being so meaningfully damaging both physically and mentally (and occasionally emotionally), [i]retaining[/i] those results would change Courage on a fundamental level—because it is a [i]gag[/i] show that has no intent of being bogged down in character development beyond the tropes that present the main trio as the archetypes they fit within (literally within the opening blurb of every episode, might I add) and nothing beyond that. Arguing that they should be taken seriously outside of that context—again, because the show must [i]always return to the status quo[/i] for the next incident—does not mean that the character has anything that would given him those powers inherently, but rather that the medium is presented in such a way that the viewer doesn't have to care about what happened before.[/list]