Here is what you would be facing.
No one is going to force you to submit the FMLA paperwork if you do not want to. However, outside of FMLA a doctor's note has no force in law. The doctor's note would not offer you any protections whatsoever. If the employer wanted to discipline you for missing two weeks of work, without the FMLA paperwork he could do so and it would be legal. With the FMLA paperwork you cannot be disciplined in any way for absences related to the medical condition in question.
On the other hand, if you do turn in the FMLA paperwork that's two weeks of FMLA you won't get back for a year, most likely. If you should need time in the next year you would only have ten weeks available, not twelve.
I can't tell you which way you should go. I don't know your employer. I don't know the likelihood that you will need additional FMLA within the next 12 months. I don't know how likely it is that your employer will discipline you for missing time without FMLA protection. All I can do is tell you what is legal in each case.