Im replacing the basement carpet with engineered hardwood and want to install heating elements under the hardwood because the basement is quite chilly year round. It is an L shaped basement and is about 800ft2.

I would prefer to only heat a "main" path to the TV area, and around the seating area to reduce the cost of heating an entire side of the finished basement that isn't used often. Is this possible? I know keeping the flooring level is critical with engineered hardwood because it is thin and tends to flex if not on a level substrate. The concrete floor is in great condition and level/smooth.

Heating below the flooring in zones makes sense to save costs. Can the heating system add less than an eighth inch thickness or less? If it is thicker you will need to shim up the adjacent flooring. An eighth inch is the specification for floor flatness variation over 6 feet for some flooring installations.

Well first of all installing hardwood on a basement floor can be a touchy situation.
Moisture emissions from the slab can be sky high. And most times it does not effect carpets because carpet can breathe.
There's a lot of electric matts out there even from a box store these days.
I'd stay away from a liquid one.
Plus you just cannot crank the heat on, it needs to be turned on slowly so as not to stress the wood out.
And I'm not sure of just a strip, typically the entire room is done because if one side or area of the room changes temp several degrees and the other does not then its going to be helter skelter with wood movement.
I'd hire a pro who has legit experience.