I agree with joed. It is
Beethoven's intention for the pianist to keep the sustain pedal constantly pressed through the 1st movement. On pianos in LvB days, 'senza sordino' would not cause a muddled sound (when playing adagio). On modern pianos, you cannot keep the pedal down, because the tones will not dissipate fast enough, no matter how slow you play. On Robert Greenberg's lectures about the LvB piano sonatas, published on CD by The Teaching Company, you can hear a sample of the way this movement sounded on a vintage piano from LvB time. In the sample he provided, the tones overlap nicely, and then the tone fades before the sound becomes jumbled.
My family and friends enjoy my interpretation of moonlight, but alas I've never played this professionally. Nonetheless, In my attempt to replicate the sound of this movement as played on the vintage piano, I mainly pedal when the bass line changes. In this movement, I keep the pedal down, and reset it only when the musical pattern or bass line changes. People listening seem to get sidetracked by the dark and haunting change in the lower register, and I suppose they do not realize that the pedal had been reset. They probably do not notice, because the change in pedaling is not very obvious if you are careful. The pedal is only depressed for a split second, and if you play legato during pedal changes, it is hardly noticeable. I think the goal is to let the notes run into each other nicely to allow that ethereal fantasia type of sound, but to avoid jumbled sound. Let your ear guide you. Josef Lhévinne says the best pedaling is when the audience doesn't realize you are using it, but with this movement that is a challenge. Try listening to Schiff, or Rubenstein play moonlight, if you haven't yet.