Why does the solubility of LiCl increase with temperature if the dissolution is exothermic?
The key to the answer lies in this comment in the answer to the linked OP:


  the dissolution of NaOH is exothermic, yet it's solubility increases with temperature.


So in short it's the opposite of #2: dissolving the first couple of molecules of $\ce{LiCl}$ is exothermic, but then if you dissolve more it become endothermic. The heat of dissolution to form an infinitely dilute solution is not the same as the heat associated with formation of a saturated solution. 

Alternately you might say that the heat of dilution of $\ce{LiCl}$ is also exothermic (see e.g. Reference 1).



Reference


YC Wu, TF Young, Journal of Research of the National Bureau of Standards Vol. 85, No.1, January- February 1980, p. 11-17.