Learn English – “would like you to” vs. “please”

Your first example is correct most of the time. When speaking about something in the future, you will be doing it. When you've made plans to do something, and you are informing another person this is the form you'd use.

The second example is usable in some situations. You're dealing here with the "future in the past" tense, if I understand it correctly. This tense is often used when presented with new information that your prior plans may influence. We'll pretend that in your example, you are speaking to someone who has asked you to write a new script. You had already planned to write a different script.

From the link:

Future in the Past is used to express the idea that in the past you thought something would happen in the future. It does not matter if you are correct or not.

When you say "Please note that I would be working on another script this month", you're indicating that you already had plans to work on another script before the conversation. In general you would use this active-voice phrasing to let the addressee know that you still intend to work on said script, either instead of the new assignment or in conjunction with it.

If you were to use the passive voice ("Please note that I would have been working on another script this month") you would usually be indicating that while you had planned to work on a new script, you will no longer be doing so.

Too complicated and long-winded is one thing, but it also doesn't sound right - I think it may be because you're repeating "you" twice, though I'm not sure. Try one of these:

  I would like to kindly ask you to check ...
  Would you be so kind as to check ...
  Could I ask you kindly to check ...
  Could I kindly ask you to check ...
  Could you please check ...

If, on the other hand, you want to ask whether somebody has already checked your file, you should go with:

Would you be so kind as to tell me if/whether you have checked …
Could you please let me know if/whether you have checked …
I would [greatly] appreciate it if you could let me know if/whether/when you have checked …