How can I safely repaint/fix peeling latex paint that was painted over lead paint?
A great many things will soften or dissolve latex paint without ravaging the underlying layer.  

The more fundamental problem is you will need a scuff-sand to get anything to bond  to the old paint.  Paint cannot bond to gloss, it needs to be microscopically rough, or visibly "flat".  

Scuff sand with a Scotchbrite "green" pad, same green/yellow sponges you use on pots and pans. You can prevent dust by doing this wet, wash contaminated water down a city sewer, and sponges in the trash.  Don't overdo it - you're not trying to level the surface, just degloss it. 

That thing you are thinking makes the new paint bind to the old: that's not what primer does, it's what scuff sanding does.  Normal primers can't do that, they also need a deglossed surface to bite into.  Primer does other stuff. 

There is no chemical that can fix the current problem.  I.E. imagine you have 4 coats 


topcoat
??? 
recent paint that isn't sticking 
glossy or contaminated old paint 


You're imagining a product at #2 that can fix the fact that #3 and #4 are delaminating.  Doesn't exist. Even at "professionals only" paint stores. All that will happen is that 1 2 3 will peel off together.



Now are you 100% sure that top layer is leaden?  Hard to believe it hasn't been painted since lead was banned. If it did last 40 years, that's the power of a good oil (alkyd) paint.  That durability has nothing to do with lead, lead was a pigment.  Luxury homes still use oil paints especially for trim.  If you can't afford top tier painters to come in, then you're the painter and nothing keeps you from doing top tier work.  

Unlike latex, super stinky oil paints are normal.  The solvents are somewhat toxic, but they stop being harmful long before they stop offending the nose.  Also they are not bioaccumulators like lead is, so once you stop using them, your body's immune system will flush out any toxins.  Paint them with good ventilation.