It's absolute rubbish (of the teacher, I presume) to insist that the second part of a quote is always capped. To check, I've just looked in my 2005 copy of New Hart's Rules.
They give the quaint sentence:
May I suggest that you have a bath before supper?
It can be either the more modern version:
"May I suggest," she said, "that you have a bath before supper?"
or the more traditional British form:
"May I suggest", she said, "that you have a bath before supper?"
The only difference is whether the first comma is within or outside the quote marks. The second part of the sentence MUST be lower capped because it is part of the same sentence, all given without punctuation.
In your sentence, however, we have the problem that the original sentence needs some sort of extra punctuation, because there is no conjugation. Thus, dividing it into two sentences is a good option:
"Help me wash the dishes," said Mum. "Then you can go out to play."
The key is the full stop after "Mum".
The only other option is trying to force a semi-colon in there somehow. But the option I've given above just looks right.