Can I safely use a separatory funnel to extract the precipitate from a solution?
Good thought, but Waylander is correct.

However, you might consider centrifugation to consolidate the ppt, then pour off the supernatant. If you don't have a centrifuge, and the quantity of liquid is not too large (100 mL or less), you could make your own, since it seems that your ppt settles fairly fast. 

Put the solution in a tough plastic bottle, cap it, and get a pretty strong string or light rope, tie the bottle SECURELY and twirl it over your head. Maybe outdoors is better for this, unless you are very careful. A few gentle swings (~20) should be enough, if it is going to work at all. I have had some success with this method a few times.

Secure tying is key, since the extra Gs will put a strain on your knots. I did something like (vertical swing, about 4 foot circle) this inside a room with 12 foot ceilings, and the plastic bottle hit the ceiling and splashed (I did not have the cap on). Fortunately, the stain was not very noticeable, but I did lose some of the material.