Most of the terms used when playing Bananagrams ("split", "bunch", "peel", and "bananas!") are all banana-themed. "Dump" is an obvious outlier - it doesn't quite fit. Some alternatives I have heard suggested are: "slip", "bad banana", "rotten banana", "bad spot". "Slip" has the one-syllable advantage, but sounds a bit too much like "split" for my taste. Maybe an exclmation like "Yuck!" would work. The "bad/overripe" theme seems appropriate, but of course "Rotten Banana" is reserved in the official instructions for someone who claims to have won but turns out to have used one or more bogus words. The word for the little gray bit at the end of the banana might work, assuming it has a name...
How about those stringy things on bananas? They are called "phloem bundles". They distribute nutrients throughout the banana. They are technically edible, but no one likes them, and people often throw them out. The analogy is perfect, and it's a short word (pronounced /FLOW em/). (Incidentally, peeling the banana from the other end (as monkeys do) is said to reduce the phloem problem, as the strings tend to stick to the peel.) The next time I elect to dump a tile, I'm going to say "Phloem!" and see how the other players react.
One final possibility (as a house rule) is simply not saying anything at all. People are generally too engrossed in the game to care.