Certainly! A test case generator for the given problem statement can be written in Python. Here's a sample implementation:

```python
import random

def generate_binary_string(length):
    return '1' + ''.join(random.choice('01') for _ in range(length - 1))

def tcgen():
    N = random.randint(1, 10**6)
    M = random.randint(1, 10**6)
    K = random.randint(1, 10**6)
    
    S = generate_binary_string(N)
    T = generate_binary_string(M)
    
    return N, M, K, S, T

# Example usage:
N, M, K, S, T = tcgen()
print(N)
print(M)
print(K)
print(S)
print(T)
```

### Explanation:
1. **generate_binary_string(length)**:
   - This function generates a binary string of the given length, ensuring that the first character is '1'.

2. **tcgen()**:
   - This function generates the values for `N`, `M`, and `K` within their respective constraints.
   - It uses the `generate_binary_string` function to create valid binary strings `S` and `T` of lengths `N` and `M`, respectively.

### Example Output:
When you run the `tcgen()` function, it will output values satisfying the constraints. Here is an example of what the output may look like:

```
435
769
123456
100011101001010110...
110101001010110101...
```

Note:
- The ellipses (`...`) indicate that the binary strings `S` and `T` will be long strings, potentially up to lengths of `10^6`.

Feel free to use and modify this generator according to your needs for generating test cases for this problem.