I can't use the toilet bucket to wash off so I always keep some paper on hand for emergencies.
I don't like to use a bucket to wash off so I always come prepared with paper in case I get a sudden urge.
in case I get a sudden urge. / in case I need to go suddenly. / in case I have a sudden bowel movement
国によってトイレの文化も随分違いますよね。理解はできます！
stomachache は胃腸の痛みを表すが、便意の腹痛の方は＝＞
a call of nature / a sudden urge / a bowel movement / bowel cramps などで言えます。
この場合には "for emergencies" / "in case of emergencies"（緊急の時のため）が一番シンプルな言い方だと思います。
桶洗い= wash (off) with a bucket
備える = prepare for; have ready
常備 = always have ready; always have on hand
アメリカ合衆国 
