I think I understand. Its the anchor line beneath the boat that would
be the problem with a fish swimming around beneath you.

One option would be to run the rope to the sand-anchor (the peg for the
beach) through the endloop of your boat. Here is how I'd think of
securing it:

Tie the end of the rope off just behind the cockpit. One way would be
to put a U-bolt in the hull, and tie the rope to it. Then put a clam
cleat back a bit, where you can reach it by turning around and reaching
behind you. The rope goes from the U-bolt, through the end loop of the
kayak, then to shore to the sand anchor.

To anchor the boat, you have to get out and put the sand anchor in,
then you paddle a bit away from shore, take up the slack in the line
that is behind you, lock the line in the clamcleat, and tuck the
extra rope into the cockpit. Now there's no extra line anywhere on the
deck or in the water, and the line to shore is coming off the extreme
stern of your boat.

Otherwise, you can just drift and let the fish drag you around. :-)

--riverman