Has the FAA ever granted a LODA for compensated instruction in a non-LSA E-AB?
You may have to contact the FAA directly to get an answer on that but it seem like an unlikely case. Since you are looking for a LODA that covers "training flights" you would need a LODA SFA 


  D.    LODA Special Flight Authorization (LODA SFA). A Flight Standards
  office normally issues a LODA in cases where frequent operations of a
  specific nature occur over a protracted period of time. However, there
  may be situations such as ferry flights, sales demonstration flights,
  and training flights where the 60-day requirement may not be feasible.
  For these situations, a Flight Standards office may issue a LODA SFA
  on a case-by-case basis for specific flights. A LODA SFA is a LODA
  authorizing an operator to accomplish short-term operations (normally
  a maximum of 30 days) in accordance with 14 CFR part 91, and can be
  issued for the following:


LODA SFA's are short term as discussed later on, specially they are typically 30 days in duration 


  ...
  
  2)    A LODA SFA is conducted in accordance with part 91 rules. The maximum duration of a LODA SFA is normally 30 days.
  
  ...


To regularly provide instruction in the aircraft you mention you would effectively have to continuously apply for a LODA SFA on a rolling basis, something the FAA likely will stop approving eventually. I would say that your use case does not really fall under the broader spirit of the regulation 


  ...
  
  LODAs to all of part 125 are issued for operational uniqueness
  
  ...


Although you could make a case if you were offering training to others specifically looking to buy that aircraft. If you are just looking to offer flight instruction the FAA is likely to tell you to go find an approved airframe and play by the rules. I cant find an exact price for a sling kit but Im seeing north of 100K for built ones, even if the kit costs half of that @ 50K you can get a whole lot of 172 or Cherokee 160 for that price and offer instruction without the hassle of special authorization (above whats normally needed).