Any method you could come up with to get some Round Up onto the green part of the fern, and not on the palm, would/should work.
I have heard or people painting it on with a paint brush. You could spray on the "easy" parts of the fern if you masked the trunk with cardboard or plastic sheets while doing it. Or, it looks as if you put some in a pail, it might be possible to dip the longer ends of the fern into the pail.
IMO - you shouldn't need to soak the entire fern. Just getting a large part of it wet should get enough chemical on there to translocate through the entire plant.
Supposedly Round Up does not affect parts of the plant that are not green - like bark, etc. But I wouldn't go about it without trying not to get it on anything other than the intended target.