Change the input source to British - PC:

Apple's version of the British keyboard layout is slightly different from the standard PC version.

Edit: I didn't notice that you were already using the British - PC input source.

You can flip the ` and \ keys (§ and ` in the Mac British keyboard layout) by using a private.xml like this with KeyRemap4MacBook:

<?xml version="1.0"?>
<root>
  <item>
    <name>test</name>
    <identifier>test</identifier>
    <autogen>__KeyToKey__ KeyCode::BACKQUOTE, KeyCode::UK_SECTION</autogen>
    <autogen>__KeyToKey__ KeyCode::UK_SECTION, KeyCode::BACKQUOTE</autogen>
  </item>
</root>