Lead encapsulation has been a safe and time-tested alternative to mitigate lead hazards
Yes, “lead poisoning” is canon in DC comics.
But luckily, numerous safety and hazardous materials organizations have studied the problem of lead exposure and have settled on some widely-considered best practices.
Lead encapsulation refers to covering up or containing a lead exposure hazard, most typically from lead-based paints, but also lead sheeting.
Lead encapsulation methods include:
Sealing the lead hazard by using encapsulating coating;
Attaching adhesive barriers to the surface of exposed lead hazard
Isolating the lead hazard by building a barrier around it.
Given the resources available to Bruce Wayne, one would think he could certainly apply lead encapsulation methods and materials to his bat-suit to protect against the hazard of lead exposure from including lead sheeting within his suit. Some examples might be a bonded surface barrier (I.e. “vulcanized” protective surface on the lead), a protective shroud or pocket within his suit that isolates contact between lead and skin, or painting the lead in the suit with an isolating coating before it’s inserted into the suit.
A risk would be penetration or destruction of the encapsulation, however, it’s more likely that whatever damaged it would be a more immediate hazard than the newly and now temporarily exposed lead. For example, blunt force trauma by someone who could shatter a mountain, explosions, projectiles, falls from tall heights, impalement by sharp weapons or objects, death rays, Omega beams, heat vision, et. al. are all arguably more immediate hazards to the suit’s wearer than limited environmental exposure to lead.
Despite these risks, Batman on occasion has lined his costume with lead:
However, because some superpowered individuals such as metahuman hitman Tommy Monaghan have the power to see through solid objects, Batman also lines the cowl with lead to protect his identity. That property is absent in The New Batman/Superman Adventures crossover "World's Finest", where Superman saw through Batman's cowl with ease.
We can presume, Bruce Wayne has either safely considered how to encapsulate his lead exposure, or has accepted the hazard with the benefits it provides to his line of action.