Ahh, yeah, a unit can be made to grant cultural influence in a square, even as the result of building an improvement. That's not even particularly hard.
But it will not necessarily work the way you want it to without some further thought. For instance... Improvements can only be built inside your cultural borders. So do you build adjacent to where you want to claim? And if so, then there would be some sort of selection process upon completing the improvement. Or do you use an action button? With a simulated timer perhaps? This can also be accomplished with some experimentation. Ok, then, what about the existing methods of cultural expansion? How do you rationalize the two?
But the short answer to your question is.. Sure, it's possiblein Python without the SDK.