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?

You’ve split the data wrong. Since class imbalance is unlikely to be a problem for your work, even if it appears to be when you use improper scoring rules, there is no need to fiddle with the data. Just split the data, perhaps stratifying to ensure the exact same ratio in both the training-sample and out-of-sample sets.
When you do this, you do not deplete your minority-class samples by artificially balancing the training data, leaving you with plenty of samples for an out-of-sample assessment (especially if you have a lot of data like you have posted is the case for your work).
An alternative to splitting the data is a bootstrap approach. Not everyone agrees with this, with an interesting debate here, and my take on it is that I am torn. However, it is worth knowing that such an approach to validation does exist.
