First, the assumption isn't that the outcome variable is continuous but rather that your residuals are normally distributed and with equal variance across your predictors. With too few levels, these assumptions are unlikely to be met. If you truly have three ordinal levels, then you'd need a multinomial (ordinal) logistic regression.

However, I wonder if there is more to the story. It sounds like you might have a binomial variable with two "trials" per subject such that the outcome is either a 0%, 50%, or 100% probability of choosing one of the categories. If so, then you need to do a repeated measures logistic regression which is done nowadays using multilevel generalized linear modeling.

Thank you very much for you answer.

Indeed, there are two trials per participant (for each combination of the independent variables levels).

I re-analysed my data using multilevel generalized linear models, The issue is that the model does not converge whenever I use a full random procedure; further, when removing some variables from the random factors, the DFs are very high.

I might be wrong, but may the use of non parametric statistics help solve the problem? (relying on rank rather than mean?).

No, non-parametric won't solve it. You may not be understanding the dfs. The random effect structure does not have to be complete because, yes, it will cause convergence issues. See https://arxiv.org/pdf/1506.04967.pdf.