Curious about this, are you using Neosporin in the ointment form or the lotion form? I know both are used to treat the same thing on people but I imagine the components are a bit different between the two and I wonder if they would work the same on plants.
Just updating: the plant is still with me! The pulb did continue to die back, but yellowing rather than going black. Unfortunately the new shoot looks dry and is probably gone too. The pulbs roots are still good though. Older pulbs are still plump and green though only one has a leaf! So I'm hopefull that it will pull through and put out new growth. I think I'll have to make sure it doesn't get too hot next summer: I think that was what started the rot in this and some Maxillarias (that didn't make it) this year.
Leave it alone. Let it heal itself. What is important is the new growth not the old bulbs. What is left of the former fat bulb will just dry off and still be a source of sugars for the new growth. The new growth is not coming from the old bulb but from the rizome as little of it there is. Leave it alone.