No, the Bucket is not overpowered. While its strengths are indeed strong, I feel it is held back from overpoweredness by its lack of range.
To its credit, the Bucket is far from a bad weapon. It has great power, being a 2-hit splat (not sure how many Defense Ups are required to lower that), and it's fairly quick. Its Burst Bombs are a good option for long-range attacks, and Inkstrikes are good for forcing enemies to move. Additionally, the Bucket retains its power and accuracy while aimed upward, allowing it to reliably splat enemies at higher elevations and over walls.
However, this thing has a couple of significant flaws. The first flaw is its range. While respectable, its range is beaten by many weapons, letting enemies with superior range mow Bucketeers down from a distance (though if you close that distance, the Bucket should either win or trade most of the time). Burst Bombs can help alleviate this weakness, but only so much. The second, less significant, more subjective flaw that I feel the Bucket has is that it deals poorly with Rollers and possibly Brushes. Any non-Dynamo Roller wins in close combat by swinging at you while you're splashing them with the Bucket (I've heard the Bucket can one-shot people at close range, but that would only upgrade this to a trade, rather than outright losing), while Dynamo Rollers outrange the Bucket, letting them exploit the first flaw. Inkbrushes win at close range (like usual) and have the SONIC SPEED to run away from you (and the Nouveau has Bubbler to prevent you from splatting them when it's ready), while the Octobrush can win or trade at close range (at least I think so, it should at least trade), has not-quite-Sonic speed to run away if it must, and can RELEASE THE KRAKEN to invalidate your attempt to splat them when their special's charged. Finally, and also subjective, the last flaw I can come up with is that it doesn't deal well with Chargers. Yes, if you can get in splash range you'll probably win, but otherwise Chargers wreck Bucket players from a distance. Burst Bombs and Inkstrikes are a way of alleviating this weakness, but only get you so far.
Overall, the Bucket has great strengths, and is probably a viable choice of weapon, but it's not overpowered (or if it is, I'm not seeing it).
But I could be wrong. I don't use the Bucket much, I prefer the Splatling, but I think I used the Bucket enough to conjecture about its usefulness.