warning. very long post ahead with lots of pics.
there are many many options for temp control. the absolute best would be to attach a glycol chiller to your in and out tube. as others have said. but funds may not allow this, and maybe you are not handy to diy it with an ac unit. thats me. im too cheap and i cant build the ac thing. i wish i could.
is this what you got? omg that thing is bad a$$!!!. my swimbo would think i absolutely went crazy. when she saw my aio, she said it looked like a small test rocket for a spacetourism start-up. is it the 7,14, or 17 gallon. smaller volumes in the fermenter cool much quicker but are more subject to temp changes. water based solutions have a high heat coefficient and the larger the volume the longer it takes to change temperature. however it will take longer to cool down and obviously require more ice/ chilling up front. if you have good insulation once you get it chilled, the larger volume should stay cooler longer. this has been my experience. small batch brewers know very well how easy it is to chill wort in an ice bath.
anyway i think you can do this without a glycol chiller. and you can do it with just a cooler and changing jugs like you intend. i think the main problem would be cold escape throught the top which doesnt look as well insulated. i would make a "hat" for it .
use a styro foam cooler . the omaha steaks one is much better than the flimsey drugstore/ beverage store beach ones. trust me its much easier to make holes in or alter. the live pet/ auarium fish shipping coolers might also work. i lined mine with a HD construction bag cause it had a leak but it might be a good idea to do this anyway. for leak insurance and i think it may add insulation by trappiing air between bag and cooler. dont drill holes you just need two notches right at the top of the cooler. the lid will create the top of the "holes". one notch needs to be just slightly bigger than the tube OD to accomodate the pump wire alongside it. just like you want a decent volume in the FV to hold temp once its cold, i find that the less water in the cooler the better because you are trying to get that cold with ice exchange . so you want it to change quick not stay warm once it "sucks" the heat out of the FV.
i fully double wrap up my fermzilla in a wool blanket.
cooler ambient temps definately require less cooling of the FV. so still if you can do this in a colder area it will be easier . i can crash mine down to 42 degrees in a 68 degree room with 5 lbs of ice in less than an hour. dont use ice to keep it cool use frozen jugs as stated above. i can keep this at 50 degrees by changing one frozen 1 gallon water jugs evrynight before i go to sleep. they are right it can be a pita to change water jugs, and if you forget it will warm up. but this works with the little extra effort. i just brewed a pilsner in here fermented at 52 degrees for two weeks then tranfered to kegs for lagering in the kegerator. i have heard of others lagering in this system at 34 degrees but it would take 5 to 10 lbs of ice daily with drainage. which is not economical for most.
the insulated tubing is a good idea also. but that bucket wouldnt keep cold at all.
this looks much better / effective for you purposes.
aparently alot of people do this sort of thing, heres more pics. :
i think this is what you want to do:
that looks like its working .
glycol chiller not needed. you can always start saving now for next black friday. lol
sorry for the long post good luck.