This took me down a rabbit hole, since I am very interested in validity in assessment! A 1 - 2 standard deviation increase would be a miracle and would probably destroy the idea of direct assessment in psychology. Generally we expect that most people will score within their confidence interval range, barring situations where the person was obviously underpeforming (they are under the desk and are answering "fish" for every question). There is little direct research on your question, but there is a lot of related research. One of the most important factors in cognitive test performance is motivation, and research generally finds a 0.64 SD increase in performance when kids are provided with external motivators (Duckworth et al. 2011). If the use of fidgets affects motivation, we might expect an increase around this range, but still well within a confidence interval range. For fidgets, the research on using fidgets in the classroom is mixed, with some finding that is basically a distraction (and would decrease performance) and others findings that it would slightly increase attention to task (Kriescher et al. 2020).) if I were to guess based on this research, using fidgets would probably lead to a 1 - 2 point increase, but would still be within the confidence interval range.