Does hybridization between native Red Mulberry and invasive White Mulberry threaten the native species?

Yes, Cindy! There are thousands of other plant species (e.g. common in Oaks [Quercus], oats [Avena], beets [Beta], mustard [Brassica x Raphenus] etc). Many people do not know that hybridization (crossing two genetically different individuals) is more common in plants than in animals/other organisms. Plant life is fascinating in that their diversity and evolution naturally is caused or stimulated by hybridization while in agriculture, plant breeding and crop improvements rely on the process of hybridization.