I tried that. according to two CFIs I tried, if you're flying as a PPL exercising Sport Pilot privileges, you must take the BFR in an LSA. It is not enough to take the review in a "sport-like" aircraft like a cub or Taylorcraft.
First of all. A medical is not required for a flight review. So the fact that you don't have one at the moment is irrelevant. The fact that you plan to fly under the sport rules in the future is irrelevant.
Second. A Cub or Taylorcraft is not a '"sport-like" aircraft - they 100%, flat out, for real, meet the definition of LSA in FAR 1.1 (Definitions). They are "Light Sport Aircraft". The method of certification for the aircraft (standard, E-AB, S-LSA, E-LSA, what forking ever) is not found in the definition of LSA. And they can be flown under the Sport pilot rules. Don't confuse aircraft certification with pilot privileges and limitations.
https://www.ecfr.gov/current/title-14/chapter-I/subchapter-A/part-1/section-1.1
Light-sport aircraft means an aircraft, other than a helicopter or powered-lift that, since its original certification, has continued to meet the following:
(1) A maximum takeoff weight of not more than -
(i) 1,320 pounds (600 kilograms) for aircraft not intended for operation on water; or
(ii) 1,430 pounds (650 kilograms) for an aircraft intended for operation on water.
(2) A maximum airspeed in level flight with maximum continuous power (VH) of not more than 120 knots CAS under standard atmospheric conditions at sea level.
(3) A maximum never-exceed speed (VNE) of not more than 120 knots CAS for a glider.
(4) A maximum stalling speed or minimum steady flight speed without the use of lift-enhancing devices (VS1) of not more than 45 knots CAS at the aircraft's maximum certificated takeoff weight and most critical center of gravity.
(5) A maximum seating capacity of no more than two persons, including the pilot.
(6) A single, reciprocating engine, if powered.
(7) A fixed or ground-adjustable propeller if a powered aircraft other than a powered glider.
(8) A fixed or feathering propeller system if a powered glider.
(9) A fixed-pitch, semi-rigid, teetering, two-blade rotor system, if a gyroplane.
(10) A nonpressurized cabin, if equipped with a cabin.
(11) Fixed landing gear, except for an aircraft intended for operation on water or a glider.
(12) Fixed or retractable landing gear, or a hull, for an aircraft intended for operation on water.
(13) Fixed or retractable landing gear for a glider.