Can a severed bicep tendon cause permanent nerve damage (numbness) in fingertips?

No: Not unless there is separate nerve damage from the same trauma.

brachial radial nerv: If the brachial plexus is affected can damage axillary radial musculocutaneous nerve may be affected also trigger points along the whole arm may become irritated which can mimic actual nerve damage. Electrodiagnostic testing will elucidate true nerve damage. Surgical repair and opysical therapy strongly considered in this case.