Here is the item again:
Because he refused to touch the serving of lima beans on his plate.
You correctly identified the fragment as a subordinate clause. Because, the subordinate conjunction that begins the clause, is the clue.
To fix this fragment, you could remove the subordinate conjunction, like this:
He refused to touch the serving of lima beans on his plate.
Or you could use Punctuation Rule 1:
Main clause + Ø + subordinate clause.
The correction would look like this:
Pierre had to wash the dishes
because he refused to touch the serving of lima
beans on his plate.
Or you could use Punctuation Rule 2:
Subordinate clause + , + main clause.
And the correction would look like this:
Because he refused to touch the serving of lima
beans on his plate, Pierre had to wash the dishes.
Enjoy having your own personal squire!