I think that this question is similar to the question should one play Bach on the piano; after all he did write his music for harpsichord.
The pedal was invented during the classical period and begun life being operated by the knees raising it (it wasn't a pedal in the same position as the current one obviously!) so Mozart wrote most of his music without the pedal being invented in the modern sense, and of course even if some pianos had these early pedals, others wouldn't have. For an authentic classical, 'as it would have been then performance' perhaps not; but you'd also have to take the engineering of your piano backwards a few hundred years too! If you feel that using the pedal enhances the music you're playing then do it; but don't overdo it otherwise your knee will get tired

; on a more serious note your playing may sound too 'inauthentic' if that makes sense (in the same way that you would be heavily criticised if it sounded overly romantic in other ways (e.g. too much rubato)).
What a detailed non-answer to the question!