As PEG-stearate is highly soluble in both water and non-polar solvents, liquid-liquid extraction might not be an efficient method for separation, but one option could be to try a biphasic solvent system such as water-ethyl acetate or water-ethyl ether. Another option could be to use a solid phase extraction method with a suitable stationary phase. However, you can try the following methods to separate the unreacted starting materials and DMAP from the carboxylated product:

1. Column chromatography: Column chromatography can be used to separate the carboxylated product from the unreacted starting materials and DMAP. You can use a suitable stationary phase such as silica gel, alumina or C18 to separate the compounds based on their polarities. You can choose the eluent based on the polarity of the compound you want to separate.

2. Solvent precipitation: You can add a nonsolvent to the reaction mixture to precipitate the carboxylated product. The nonsolvent can be an organic solvent such as diethyl ether or hexane. The carboxylated product should precipitate out of the solution and can be collected by filtration. The unreacted starting materials and DMAP will remain in solution and can be separated using column chromatography or other suitable methods.

3. Acid precipitation: You can add an acid such as hydrochloric acid or acetic acid to the reaction mixture to protonate the carboxylic acid groups of the carboxylated product. This will make the product insoluble in the reaction mixture and it will precipitate out. The unreacted starting materials and DMAP will remain in the solution and can be separated using column chromatography or other suitable methods.

4. Distillation: If the boiling points of the compounds are significantly different, distillation can be used to separate them. This method involves heating the mixture to vaporize the compounds and then condensing the vapors to collect them separately.

The choice of method will depend on the specific properties of the compounds and the amount of material available for separation. It is recommended to perform a small-scale trial to determine the most effective method before scaling up the process.