It should be okay for you to apply for ESTA, because the visa you hold is not a B-1 or B-2 visa. If you were to travel to the United States for purposes of tourism, or business purposes unrelated to the visa you now hold, you would in fact be required to apply for ESTA or for a B visa.
(I'm not sure, however, what will happen if you can check in using your ESTA, and subsequently enter using the visa.)
For example, at https://travel.state.gov/content/visas/en/employment/treaty.html, you can see that for an E-2 visa "you must be coming to the United States to develop and direct the enterprise."
See also at https://www.cbp.gov/travel/international-visitors/frequently-asked-questions-about-visa-waiver-program-vwp-and-electronic-system-travel:
Q: What if a traveler has a current, valid visa?
A: Individuals who possess a valid visa will still be able to travel to the United States on that visa for the purpose for which it was issued. Individuals traveling on valid visas are not required to apply for an ESTA authorization.
This implies that you can apply for ESTA to travel to the United States for a purpose other than one for which your valid visa was issued.
You also express concern about whether you might have trouble when entering the United States, since you will have been indicated on the carrier's passenger list as a VWP traveler, but you will be presenting a visa at entry. I doubt that you will have trouble because of it. I can think of two reasons you might:
- Applying for ESTA with no intention of using the VWP
Title 8 of the Code of Federal Regulations, section 217.5 (a) says
Each nonimmigrant alien intending to travel by air or sea to the United States under the Visa Waiver Program (VWP) must, within the time specified in paragraph (b) of this section, receive a travel authorization...
You might therefore run into trouble because you don't actually intend to travel to the US under the VWP. But it doesn't actually say that those who don't intend to travel cannot receive the authorization. Furthermore, CBP notes that people can apply for ESTA even if they don't have definite travel plans.
- Checking in with ESTA and entering with a visa
Here, I can only speculate, but I guess that the carrier is more likely to get in trouble for this than you are. You could approach the ground staff at the airport to correct your record with information about your visa, but this would defeat your apparent purpose of getting on the plane quickly and easily.