The validation set includes few positive labels I'm training a classifer on an unbalanced dataset. The test dataset's positive proportion is 0.02%.
For that reason, the validation data set labels proportions are the same. Because the validation set size is much smaller than the test dataset, it contains less than ten positive labels. The test set includes 25 positive labels. I tune the model hyperparameters by using the F-beta score.
I'm not sure that a sample with less than ten positive labels, is a valid sample for tuning and evaluating the classifer. Indeed, the classifer has terrible results when applied to the validation and test sets.
Since the training set is more balanced from the validation and test sets, I can move positive labels from the train set to the validation set (and test set). However, in that way, they will not represent the real data.
What do you recommend me to do?

Having a balanced training set is important as the model may otherwise learn to be biased towards the negative class. If it is not possible to have a balanced training set then I suggest you look into adjusting the weight each class has on the error criteria. This can be done in sklearn by using the class_weight model parameter. In turn, giving the model a much larger penalty during training for misclassifying positive examples to make up for the lack of positive examples in the training set.
As for the validation set, 10 positive examples is quite small. Try to increase that to 20 if it's possible. However, having an unbalanced validation and test set is not as much of an issue. You can look at the confusion matrix or other metrics derived from the confusion matrix to validate the performance of the model - https://towardsdatascience.com/performance-metrics-confusion-matrix-precision-recall-and-f1-score-a8fe076a2262.
Furthermore, have you looked into augmenting the data?
