Digging them is the only really safe method I can think of. Sahara will probably work as will diesel oil.
A weed wrench, large weed popper or puller will allow you to extract the entire plant. *Several sizes of various weed pullers are available depending on the size of your palmettos.

Would do this except the ones in question are growing all up thru the
roots of large southern pines. There is no way to pull them up as
they are completely intertwined with the pine tree roots. I can try
pulling up the hearts but prior to cutting down to the ground, some of
these palmettos were about the size of a decent sized VW beetle :O/

Anything like "stump killer" or anything that can be pour carefully
into this heart and kill the palmetto without seeping into the
surrounding ground and harming the pine trees?