Are there any objections to using the same (unlabelled) data for pre-training of a BERT-Based model and the downstream task?
Not at all. A recent ACL paper by AllenAI even says this is the best way. They recommend continuing pre-training on the task data and claim that it reduces the problems caused by domain mismatch. So, if you train the model on the in-domain data from the very beginning, it is probably a good thing given you have enough data for that.