Under the circumstances I'd really recommend AGAINST an undergravel filter. (Full disclosure: I don't generally like them all that much anyhow, even under the best of conditions). Biggest reason being, mostly it will pull all the fish poo and uneaten food down into the gravel and underneath, where you can't as easily remove it by vacuuming, and it will take AGES for enough bacteria to build up to process it properly.
Why not just get a regular ol' hang-off-the-back-of-the-tank power filter. Just as cheap, MUCH easier to set up (does not require totally wrecking apart the tank!), better with live plants than an undergravel filter is (plant roots can restrict flow thru it), and will IMHO do a better job of allowing you to manage high nutrient loads.
Then vacuum the tank every week or two, doing a small water change to replace the volume lost in the vacuuming. Make sure to stir the gravel around a bit. As you say, goldfish are MESSY.
You may be feeding them too much. All they need is a REAL LITTLE pinch of food, no more than once a day. Honest.
Finally, I would HIGHLY recommend getting a test kit for nitrites and possibly nitrates. So you can see how bad your water is at any one time
and whether you need to do more or not.
BTW, you mention live plants. How much lighting do you have for them. I ask because live plants in insufficient light (and most of them really do need a WHALE of a lot of light) will start dying back and actually *adding* nutrients back to the water, rather than what you probably hoped for which was that they'd sequester nutrients *out of* the water.
If you still find yourself with real bad cloudy-water high-N problems that a decent filter and regular vacuuming and suchlike won't cure, there are products ont he market that you can put into the filter or tank that will chemically remove some of the ammonia/nitrite/nitrates from the water. I don't normally like them one bit, but if you are stuck with too much goldfish in too little tank there might be some use for them. They are not real cheap but they won't kill ya either. Try all the other management measures first though.
GOod luck,
Pat