I have a degree in chemistry. This is probably fine unless used constantly or in excess, or if you water is too hard. Powder dish detergent contains sodium carbonate, which is more basic than sodium bicarbonate (baking soda). Bicarb is less soluble and thus acts as a very mild abrasive, but powder detergents usually are similarly abrasive, though I can't find any hard sources saying bicarb is completely safe.

Biggest issue would be hard water creating gunk buildup in the plumbing. Run the "good stuff" every now and then with drying agent and that should clear it out.

If you swap bicarb with sodium carbonate (washing soda) you literally have dishwasher detergent that might be a bit foamier.