Sure, I can help you with that. Below is a test case generator for the given problem statement:

```python
import random
import string

def generate_random_name():
    # Generate a random name consisting of numbers and uppercase letters
    length = random.randint(1, 100)
    return ''.join(random.choices(string.ascii_uppercase + string.digits, k=length))

def tcgen():
    # Number of members in the programming research club
    N = random.randint(1, 1000)

    # List of names
    names = []
    num_e869120 = random.randint(0, N)
    
    # Add the specified number of E869120 copies
    for _ in range(num_e869120):
        names.append("E869120")
    
    # Fill the rest with random names
    for _ in range(N - num_e869120):
        names.append(generate_random_name())
    
    # Shuffle the list to make it more random
    random.shuffle(names)
    
    return N, names

# Example usage:
if __name__ == "__main__":
    N, names = tcgen()
    print(N)
    for name in names:
        print(name)
```

### Explanation

1. **generate_random_name**: This helper function generates a random name string consisting of numbers and uppercase letters with length between 1 and 100 characters.
2. **tcgen**:
   - It first selects a random number \(N\) of members in the club within the range [1, 1000].
   - It then decides how many copies of "E869120" should be in the list.
   - It adds the specified number of "E869120" and fills the rest with random names.
   - The list of names is shuffled to ensure randomness of the test cases.

This will generate test cases that thoroughly test the problem statement by including both cases where multiple instances of "E869120" exist and where none exist.