Thank you for your question and patience, I’m Tom and I’ll try to help you.
You can apply for naturalisation as a UK citizen provided that you meet the criteria on the following page:
Please note I assume you are not married to a UK citizen, but let me know if you are.
If you meet the criteria on the above link then you can naturalise. However, it is expensive (£1236) and it can take up to 6 months to determine (sometimes over).
I was going to post the link that you posted. If you are not minded to naturalise then you would have to apply to transfer to a NTL and receive a BRP in replacement of your current stamp endorsed in your passport.
If you submit your application to naturalise to the home office then they would simply determine the application, they would not write to you in the meantime for the purpose of confirming to your employer unfortunately.
I would decide which is preferable to you and then check with your employer, but it’s likely they’ll prefer for you to apply for a BRP rather than wait the 6 months to naturalise.
My goal is to provide you with a good service. If you feel you have received anything less, please reply back as I am happy to address follow-up issues specifically relating to your question.