To say that the rich man must have been in purgatory because love cannot exist in hell is a conclusion based on an unsupportable premise. The Church does not teach that those in hell are bereft of all kinds of love. It is true that supernatural love of God cannot exist in hell, but a disordered love is involved in every mortal sin, and this perverse loving will remain.
What may appear as the rich man’s love and concern for his brothers may in fact be nothing more than self-interest. Thomas Aquinas asserted that the rich man knew that if his kin were damned his own suffering would increase. “[The damned’s] punishment would be greater if all their kindred were damned, and others saved, than if some of their kindred were saved. For this reason the rich man prayed that his brethren might be warded from damnation: for he knew that some are guarded therefrom" (ST Supp.–III:98:4 ad 1).
Also according to Aquinas, the damned are consumed with envy for those who attain glory, even for their own kin, though perhaps to a lesser degree.
God does not purge people of all types of love before they enter hell, but in hell the separation from God and his divine love is accomplished forever, making supernatural love, or charity, impossible. Love of evil, however, remains.