Negative back at you. You can take an E-LSA out of the definition of a light sport aircraft. It will then require a PPL to fly it. The A/W doesn't change. Once the airframe has had a C/S prop, it can't ever be flown as a light sport aircraft again. Believe what you wish. I won't distract from the core intent of the thread any further.
Ok, then for everyone else.....
For an E-LSA to receive its airworthiness cert., the kit manufacturer and the builder must both certify on a Form 8130-15 that the airplane kit and assembly processes used, built and assembled the airplane to be a copy of the original S-LSA aircraft that was previously certified to meet the LSA consensus standard (met all of the requirements of an LSA aircraft such as stall speed, max speed, and a lot more).
Certifying that it meets all of those requirements is what makes it a Light Sport Aircraft.
Because it is experiential, it can be modified to what ever degree the owner wishes..... as long as no modification causes a performance change that makes it no longer meet the requirements for a Light Sport Aircraft.
If a modification did make it perform outside the requirements, its airworthiness certificate is no longer valid.
The problem is that there is no LSA police. You are on your honor that the airplane you are flying, regardless of what you have done to it, still meets all of the LSA requirements. This has nothing to do with what level of certification the pilot has.
The bottom line is if you are still flying with an E-LSA airworthiness certificate, it is still an LSA aircraft. It can't be an LSA aircraft if it doesn't meet the LSA performance requirements.
E-AB is totally different
An E-AB that was originally certified as meeting the LSA requirements (such as an E-AB RV-12) does not have this burden, because meeting LSA requirements is
Not a qualifier for receiving an E-AB airworthiness. It just happened to be an E-AB aircraft that the builder claims meets the requirements which then allows a sport pilot to fly it. If modifications are made that would make it no longer meet the light sport requirements, its certification is still valid (because they weren't a requirement) but now a Sport Pilot can no longer fly it.