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Battler's particular degree of incompetence in that context is not relevant in any factual or evidentiary sense, only in terms of how he is performing.
And by the way, a person having an abstract quality such as competence actually cannot be verified in reality. It's unfalsifiable; you can only demonstrate whether they competently handled a particular situation. The fact that a person failed to assemble a puzzle does not actually prove that they are incapable of solving any puzzle or indeed that they are even incapable of solving that puzzle. They could be feigning incompetence or just not interested. That one puzzle could just be really hard for them. Any assessment of their overall competence derived from that would solely be the opinion of the individual doing the assessment.
That's incredibly pedantic, but a spade's a spade.
Competence is not a binary state. Additionally, factual evidence suggests Battler possesses some measure of competence. It's simply more reasonable to believe Beatrice was joking or expressing an opinion than making a statement of fact. A statement of fact in that situation which is indisputable would be "The logic you are using cannot possibly lead to the correct solution." This is a fact statement properly tailored to the situation and cannot reasonably be construed as her opinion (it could potentially be viewed as such, but at that point the author would lose our confidence).
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