I haven't got any experience myself but have done a lot of reading about duckponics (although I myself much prefer the ring 'quaquaponics' has to it

) of late, and would also be interested in hearing others input on this subject, although I'm not sure your system really qualifies as the water isn't recirculated, if we want to get all technical about things
Have you tried anything out yet or made any updates? I'm sure you have already encountered the problems that I can foresee with your system, whether trying either irrigation method or just emptying the tub for your ladies! That being the need for some sort of filter for all the sludge/feathers/assorted crap they've dragged in with them.
As for the irrigation itself, from the opinion of someone who's never attempted anything of the sort, if the water is dirty enough to block emitters then would it not also block holes drilled in the irrigation pipe? I would also assume that the bigger the holes you drill the more pressure you would lose, so simply increasing the diameter of the holes to stop them blocking wouldn't work?
I had to google what emitters were (that's how little knowledge I have so please forgive me if I'm talking nonsense) and see that there are 'pressure compensating' emitters for various flow rates, so would that be what you were recommended? I think if you could use them, if needed, no improve the flow you get, and something to filter the larger of the debris from the water then it could work great! How far apart are the beds and the bath? would it be possible to add a gravel bed along the water's journey to your beds? This could serve as part of a filter system and maybe also play host to some beneficial bacteria that would help to kick start the process of breaking stuff down so it's not all done in your beds and the plants can benefit sooner.
Could this second, closer reservoir help create more pressure, or would you lose pressure at the pipes if it has less distance to travel?
Emma