Label A records B times or label A*B records
In my opinion there's no correct answer between the two choices, and you are right about the arguments. 

I would argue for a balanced compromise between the two: have a proportion of A (say 20%) labelled by multiple annotators (say at least 3), but the rest of A can be labelled by a single annotator. You could refine the proportion/number of annotators progressively based on the results: for instance if after the first batch one observes that annotators disagreement is very high, then it might be useful to increase the two parameters. This way you can still evaluate the level of disagreement and it effect on performance, while also maximizing the amount of data annotated.

An alternative is to use some form of active learning. For example, first use method 2 (each instance labelled once), then use cross-validation to tran a model and apply it to the corresponding test set. At the end of the process, the misclassified instances are the "hard ones" which need to be re-annotated by multiple annotators. This kind of process can be used iteratively to identify the most "ambiguous" instances.