What organic solvents are suitable to use with potassium permanganate?
It sounds like you need a solvent that 1) does not react with KMnO4 and 2) is immiscible with water. 

As @Janice DelMar said in her comment, some of the lower-chlorinated hydrocarbons (methylene chloride, 1,2-dichloroethane, 1,1,2-trichloroethane) should work. If you can live with a high boiling point solvent, you might consider chlorobenzene.

Other solvents that fit these criteria include most alkanes (pentane, hexane, cyclohexane, petroleum ether, etc.), benzene (but not toluene), and diethyl ether.

A quick test to determine compatibility of your solvent with KMnO4 involves a TLC stain. Make (or borrow) a batch of KMnO4 stain as described here. Dip a TLC plate into your solvent and then dip it into the stain with minimal drying between. If the plate turns brown (MnO2), then your solvent was oxidized and is no good. If the plate stays purple, then your solvent is resistant.