How about the outdoor Shiba Koen pool? (AKA Aqua field)
Minimal swimwear and a (small) children's area with elephant slide along with the 50m pool.
Cost is 400yen and you can get the Toei 88 bus there from Shibuya station.
http://www.kotsu.metro.tokyo.jp/bus/keitou/2604.html
Timetable from Shibuya
http://tobus.jp/cgi-bin/pctimetable....06360000&hcd=0
Apparently, you don't need to wear swim caps here.