Can a desolder pump aerosolize lead particulates?
If you're worried about lead, I would switch to lead-free solder. It takes some getting used to, and you may need a hotter iron, but it works fine.

I hand solder 0.5mm pitch parts regularly using Sn99.3Cu0.7 solder.

It's better for the environment, in addition to potentially being healthier for you.

A simpler change you could make is to use flux and a desoldering braid instead of the solder sucker. The braid has some of its own flux, but some extra can really "wet" the solder to get it out of through holes and crevices. Some of the cheap desoldering braids work terribly, so I'd recommend springing for the good stuff. Get something with a good rating on amazon. And pick up a highly rated flux at the same time.