What method could I use to burn a small volume of liquid octane in a lab experiment

(Ultimately to find the enthalpy change of the combustion of the octane)

The best way would be to use what is called a bomb calorimeter. You have a small pressure vessel you put the octane in. You charge the vessel with O2 and put it in a water bath with a known amount of water. A small wire ignites the octane and you measure the temperature rise of the water and calculate the heat released. This is rather elaborate appartus and is used on commeercial labs, you may not have one available to you. BTW, I did this experiment in college chemistry, our lab had a number of the devices so we all could use them.

Otherwise, I would not attempt this. Octane is similar to gasoline, it is like you would be burning gasoline. If you burn in it a small cup, like in the peanut experiment, you would lose much of the heat to the atmosphere and would also lose some of the octane as vapor. Plus there is the safety factor.