Well, I tried this with my PSR-E433, and if the E403 works in a similar manner, this won't do it, either. First of all, to induce a no-chord situation, I had to play 4 chromatic notes together -- like C, C#, D, and D#. If I just played 4 scale notes together (C, D, E, and F), the keyboard style still did something with it (C11? Dm9? Not sure...).
However, since the 4 chromatic notes played together induced a no-chord situation, the only bass notes I got were the root note -- in this case, C -- played over and over or in octaves, instead of any kind of walking bass.
So, it looks like Michael so far has the best solution... If the E403 can accept an external style, then create a style that includes just drums and bass. Maybe take an existing swing style with walking bass, and just edit out everything except the drums and bass -- essentially, what I originally suggested, but to remove the unneeded tracks directly from the style itself, instead of muting them on the keyboard.
Now, in more recent years, I learned that on my E433, I can have auto-accompaniment engaged AND split-keyboard engaged at the same time, and if I do this and play an actual chord with my left hand, then I get both the style's auto-accompaniment as well as the chord I'm playing using the split voice. If the E403 has this same capability, then that gives added flexibility with this option to create a style with nothing but drums and bass. For example, after creating and loading such a style, you could set up, say, a piano sound as the split voice (you may want to raise it an octave or two so it is not "muddy" sounding). If you just want to hear drums and walking bass, turn split voice off (but of course, keep auto-accompaniment on). But if you want to add the piano to your background, turn split voice on, and then play chords in rhythm with your left hand, and you should hear the drums, bass, and your piano playing.
This may even work with easy-play chords, as well. In other words, if you have the style and auto-accompaniment on, as well as the split-voice turned on, and you just play a single C note with your left hand in the lower part of the keyboard, the keyboard may actually play a full C major chord using the split voice. I'd have to double check this on my keyboard to see if that works.