@JimD I like your screen idea, and may try it myself. I've just been dumping the whole mess from the target box into a large storage bin (the big black ones from Costco) and shaking it to get the lead to the bottom and the lighter stuff on top. I then hit it with a blast of air from a leaf blower (the battery ones make this easier, as I work my way down in layers, and of course do it outside) and this gets all the cardboard an paper bits to blow away. Then I scoop out the good clean mulch and put it back in the target box.
I simply repeat this a few times working down in layers until I get to the last layer of mostly pellets but also some small rubber mulch bits mixed in. After a good blast of air, I pour the contents into a five gallon pail and then switch to the shop vac to get out the rubber mulch leaving the pellets behind. I have found that what works best is to use an extension tube on the end of the shop vac hose, but to hold it in place a with a small gap to leave a leak path to reduce the suction some at the working end of the tube, and then I can put the tube right on top of the pellet and mulch mix and the pellets end up being too heavy to rise up but the mulch does go up and is removed. Someday I may make a "tool" out of PVC pipe with a hole in it, but the gap method works great. I typically have to shake the mix around a bit and do it a few times, but the result is a collection of really clean used pellets.
I've been keeping my lead with the thought of melting it down and casting it some day, but have not yet as I pretty much just shoot .22 and don't see much point. I'll probably end up selling it for scrap or giving it away some day. The stuff from the rubber mulch trap is clean but the stuff from earlier in the duct seal trap is messier.
Anyways, you could do a mix of things and do what you do now, but add a blast of air and then the vacuum to the results of the sort to clean it up a bit more before melting. Here is what my results of the method above typically look like: