According to my research most ASW weapons like torpedoes, mines and missiles rely mostly on acoustic (sonar/radar) targeting, and possibly MAD (magnetic anomaly detection) or visual sighting.
Well, radar is not acoustic, it's electromagnetic. But in any case, as you point out, there are many different detection mechanisms and so they would try them all until one of them happens to work.
I suppose you could declare that the monster is invisible to every single detection method. If it is at least visible, and goes to shallow water, you could find it with spotter planes. Then simply drop depth charges in the area and see if a bunch red stuff floats up... Assuming the blood and guts aren't also invisible.
If it is even invisible to sight, okay, no big deal. Even if it's acoustically invisible, things that move fast in water lead to cavitation which is sound generated by water itself. Not much the monster can do to hide that, unless it has a way of moving fast without cavitating. If it does, no sane navy would destroy it under any circumstances - it would be worth its weight in gold ;-).
The one possible way around cavitation is decoys as you mention... But then this is just the one monster, and kind of an important target. Can't they just shoot torpedoes or depth charges at every decoy? How many decoys does it make? At some point you can triangulate the monster's true position simply from statistical analysis of the decoy locations.
In any case, locating the monster is super easy. You just tag it with some kind of transmitter. When it's detected in an area, drop a bunch steaks with an active transponder (sound, radio, anything) inside, so the monster swallows them. Or if it only likes boats, send some unmanned drone ships with the same. If it doesn't like the drones, well, no pain no gain - just mandate transponders on all civilian ships (or maybe you can simply home your torpedos/missiles/whatever on the existing transmitters the civvie boats have) so that no matter what it eats, it will swallow one.
Granted, if you can make it swallow things, why not just feed it a bomb? I suppose you want to capture it intact, for the stealth sub technology... In any case, if feeding it transmitters doesn't work for some reason (magically digests everything?) then you can station riflemen on every ship or convoy that shoot a transmitting dart when they spot the monster. This would work even if it's invisible - you'll see the ship being eaten, so just shoot at the air around it.
Depending on the size, speed and habits of the monster you could deploy large nets. These nets could have some kind of transmitter which would once again enable detection. Not very eco-friendly -- but it sounds like there's only the one monster, and a comparatively small piece of ocean you would disturb.
Lastly, there is always the overwhelming firepower route. You could have patrols by planes, helicopters and fast ships in the area it is spotted, even escort civvie convoys with warships. As soon as the monster attacks, immediately fire cannons, cruise missiles and the like. If it's moving around too much, you can just target the ship being eaten, depending on how long that takes. You can also just launch several speculative ones in directions it's likely to go next. Since there's only the one monster, being efficient with munitions is presumably not a concern. And of course, there's always the literal nuclear option. Again, not very nice to the fishies, but they used to do it all the time for testing, and it's really just the one time. What's one more underwater nuke?
But the more I think about it, the more I wonder, why would the military want to destroy it? If it's giving them so much trouble, then it's extremely valuable for the development of submarine technology. I imagine they would try to capture it alive by luring it into some cove, or at least kill it without destroying the carcass. While this delicate process is underway, they could deploy many unmanned decoy ships to minimize the danger to civilians.