Why can't we say "पीटर सर्दी में नहीं खेलता है।" ?
This should be accepted. You can report if you come across the sentence again.
Why is there no है at the end of this sentence?
The है is optional in a negative sentence that uses नहीं but it is not wrong to include it.
Is it wrong to have है at the end of this one?
Khelta nahi is a perfectly good Hindi syntax
'खेलता नहीं' is fine when you want to place emphasis on the negation.
However, when using the inverted word order, the है is usually not omitted. So, 'सर्दी में पीटर खेलता नहीं है' should have been accepted.