How you go about getting the poison ivy out of the saw palmetto will depend to a great extent on your sensitivity to the poison ivy. Personally, I'm not sensitive, so I would just rip it out. I wouldn't worry about injuring the saw palmetto (they're pretty tough) so much as I would worry about the palmetto injuring me!
I don't know how the Saw Palmetto (Serenoa repens) would react, but our common (on the east coast of Florida) Dwarf Palmetto (Sabal minor) is not at all impressed with Roundup. Roundup kills poison ivy, so maybe another member has tried to kill Saw Palmetto with Roundup and can advise if your plant is likely to be injured by a little overspray. That might be a viable option if you are really sensitive to Poison Ivy.