The size of the document produced by exceed the BSON size limit, I believe that
would not make any difference.
The problem with $facet is that it produce only 1 document. In another thread about $facet I came up with an hack that uses $unionWith rather than $facet to get the same answer but in a different format. May be you may adapt this solution to your problem.
You could also share your $facet and we could propose other alternatives.