I would not go for a mixer for this recording task.
One alternative would be to consider a stand-alone recorder, and the Zoom H4N would work for you. It has 4 tracks with 2 built-in mics and 2 XLR inputs for external mic or line inputs. It records to flash memory cards, and can plug into a computer USB port for transfer of the recorded tracks for editing.
The other alternative is if you have a computer to use for recording (assuming that your mentioning an interface means that you do have), you could get a USB interface box such as the Lexicon I·ONIX U42S or the Presonus AudioBox 44VSL. Both of these units have 4 mic input channels, are reasonable quality and come well within your budget.