What are the advantages or disadvantages to billing a commercial insurance company our negotiated rate according to our fee schedule with them, as opposed to billing our regular, higher rate?

Charging the payer only its allowable is called "netting the A/R" or "netting the charges" and it creates three problems:

It misrepresents your usual and customary fee, meaning the insurance carriers will start loading in lower fees.

It might cause you to miss an opportunity for a higher payment if you guess at the allowable or if the carrier has changed it.

Your gross charges become meaningless, so reporting on this data point is difficult.

Some practices do indeed net the charge, but it's rare.