Here is the fragment 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!