'Don't let me down' sounds like John is playing those chords with both pickups on, probably with the volume at 9 or 8, a little bit of gain (but remember, P90's break up all on their own for the most part, so very very little gain), and those tones dialed way down. If I really want to emulate the sound of the record I usually just add mud! Dial the tone waaayyyyy back to 1 or 2 on the bridge, or 3 or 4 on the neck and either play just the bridge or bridge and neck. Instead, when I'm practicing with the band, I go right into Get Back with my bridge pickup's tone fully open at 10.
I'm usually not too satisfied if I directly emulate the sound of John's Casino playing on that album. The Casino sounds on that album were the muddy, muffled guitar sounds behind all of it. Like a lot of Casino owners, I went to the later Beatle years to dial in some sounds and was surprised that the crisp lead sounds were, for the most part, George with his Rosecaster or Les Paul. Its tough to do Let it Be without there being any confusion.