Liszt's Églogue, from the first part of Anées de Pèlerinage, contains several instances of a theme, first occurring between bars 3 and 6, in which a melody is played over a (much lower) bass note. The chords are arpeggiated, but how would one go about sustaining the bass note while not affecting the melody? I have tried using the sostenuto pedal, but it is very hard to catch the bass note while also keeping a light texture. Any suggestions?
If you can stretch a tenth, you can sustain all the notes with your fingers if you really want to. Play the Db in bar 3 with your right hand, etc.
Otherwise, just use the sustain pedal (possibly with a half-pedal half way through each bar) and don't fuss too much over it. If the piece as a whole sounds convincing, nobody will care if the notes literally sustain for exactly the written durations.
The sostenuto pedal was first exhibited in public in 1844, but not patented until 1874, in both France and the USA. Steinway in the USA was the most "enthusiastic" early adopter. Anées de Pèlerinage were composed in 1848-55. Liszt may have seen the device, but on his virtuoso tours around Europe it's unlikely that he played many (or even any) instruments with one.
If you want to use the sostenuto pedal, you just have to make the first "arpeggio" slow enough to use it. They are often somewhat temperamental and lethargic animals. The bass notes have the same pitch for 4 bars, so you only have to "fire" it once.