```python
def line_tag(body: str) -> str:
    """Compute the tag of a body as the sum of its character codes modulo 256,
    rendered as exactly two lowercase hexadecimal digits, zero-padded."""
    if not isinstance(body, str):
        raise ValueError("Body must be a string")
    tag = sum(ord(c) for c in body) % 256
    return f"{tag:02x}"

def check_line(line: str) -> str:
    """Verify a tagged line and return its body."""
    if not isinstance(line, str):
        raise ValueError("Line must be a string")
    if len(line) < 3:
        raise ValueError("Line must be at least three characters long")
    if line[-3] != '~':
        raise ValueError("Third-from-last character must be a tilde")
    body = line[:-3]
    expected_tag = line_tag(body)
    actual_tag = line[-2:]
    if expected_tag != actual_tag:
        raise ValueError("Tag does not match the body")
    return body
```