I've heard others ponder this sort of thing and have usually disregarded it.
AP as we know it is a closed loop.
But now I'm pondering too... I doubt it's worth pursuing but have a few thoughts.
(Rambling thoughts follow, don't trust that any of them are helpful or correct

).
- Firstly your local water authority would probably not want you to do it, due to water rights and issues from 'polluted' (fertilised) water effluent being discharged into the local water system. I'm sure this would be a fairly large impediment to trying this out.
- thoughts- Regulations aside, perhaps if your beds were large enough compared to your feed ration that you could in theory return water without significant nutrient levels. But to do so the plants in the last portion of the beds would have to be nutrient deficient to even come close to achieving this.
Grass beds or similar as the final beds to suck up excess nutrient might be an option since deficiencies in a 'compost crop' wouldn't be a real concern. This would be a little bit of a waste of nutrients and probably painful when the roots become an issue.
- Your spring water would have to have constant parameters that are safe for fish and plants.
- I suspect that you'd need a lower flow rate than our recirculation systems.
In 'our' AP systems it's the constant circulation of the FT water through the media that achieves the biofiltration of the ammonia. You'd need to biofilter the water before it hits your beds (maybe?) so that the first section of the bed receives Nitrates. And perhaps the nitrification process couldn't be completed before becoming effluent water. Wasting the expensive fish sourced fertiliser and polluting the local waterway.
The only reason this has grabbed my imagination this time is it would be a non-mechanical/electric kind of system. As a mental exercise I like the thought (apologies for playing devil's advocate and possibly encouraging you to do the impractical... I'd hate to do that, the systems that are 'standard' here have AP down to a science and work very well, going outside of the norm too far would probably be a painful mistake).
It's possibly/probably not viable or practical but maybe (just maybe) with a slow enough rate of water flow into the system it might be plausible (heaps of caveats there!).
A very big maybe though. You'd be taking on a fair risk and flying a little blind trying to pull it together.
I wouldn't advise doing it but it does intrigue me. I really wouldn't advise it... probably should have kept my silly ponderings to myself... (feel free to flame me up people, I think I've been a bad boy).