So I've got this Rickenbacker 4003 copy which has average cheapo pickups in it... they actually sound OK and do the job well however I've noticed that the low E string is a lot quieter compared to the other 3 strings. Looking closely, the pickups are actually mounted slightly off-centre to the strings, but not by much. I did a test bending the string down and pushing the pickup as far across as I could but it didn't seem to do a massive amount to fix the volume level, so I'm not sure it's entirely because of pickup placement. I've tried raising the pickup on one side but it makes the A string louder as well, which I don't want.

Could there be a problem with the pickup magnet or something? I don't know a lot about how pickups are made but I would think if there's going to be a problem with volume it'd happen for all strings not just one!
What amp are you using? If your amp's speakers are too small and wattage too low, it simply might not have what it takes to reproduce that low E note.
There's no such thing; there never was. Where I am going you cannot follow me now.
theogonia777 At first I used my Fender FM212 guitar amp (obviously not ideal - it was just there and set up already), I thought the same thing so plugged into my Ampeg Micro VR 200 watt running through 1200 watt Ashdown 810 fridge cab... still not great :/
Looks like a speaker cab problem, not a bass PU problem. The lowest note on your bass is 41 Hz. The lowest useable frequency from your Ashdown is 70Hz so the lowest 1/2 octave is not reproduced well with this cab. The Frontman 212 is useless for testing bass response.
Ashdown specs:

A Rick 4000 series bass is known for snappy tone, not big fat bottom like a P-bass. How do your other bass instruments sound through that cab?
"Your sound is in your hands as much as anything. It's the way you pick, and the way you hold the guitar, more than it is the amp or the guitar you use." -- Stevie Ray Vaughan

"Anybody can play. The note is only 20 percent. The attitude of the motherfucker who plays it is 80 percent." -- Miles Davis

Guthrie on tone: https://www.youtube.com/watch?v=zmohdG9lLqY
It takes quite a bit of power to reproduce low end bass -- that 200W amp is struggling. But if your other basses aren't having any problems with low end, then...
It's possible that the pickup is having issues as well. Sometimes the pickup cover on a cheap bass hides a pickup that wasn't really designed for bass.
Light, floppy low E string? Dirty ol' E string needing replacement? Short scale bass?
Just referred to this but didnt help me... but i figured out my own issue...
My solve (for Pbass configuration) was to flip the pickups, the split pickups may be wired differently for tone.

Try switching them.