NavyRecognition reports the Navy has selected the sonar for their Anti-Submarine Warfare Continuous Trail Unmanned Vessel.(ACTUV). Since this vessel is quite small at 140 tons and 132 feet long, there is a good chance the sonar would fit nicely on cutters as small as the Webber class, and certainly on the Offshore Patrol Cutter.
“MS3 (Modular Scalable Sonar System–Chuck) marks an improvement in the performance and reliability of proven sonar technologies. It performs active and passive search and tracking; incoming torpedo warning; and small-object avoidance for safer navigation.”
Maybe it is time to “Bring Back the Coast Guard ASW Mission.”
Chuck Hill's CG Blog 
It is interesting to wonder just how good modern sonar sets in escorts and submarines must be when you consider the astounding performance delivered by dipping sets used in helicopters and sonobuoys from MPA.
It was good to hear that we in the UK will be soon getting back into the MPA game with the procurement of 9 P-8. Like with most defence buys more would have been nice but it is a start.
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Another sonar, this one going on the Finish Hamina class missile boats. The Hamina class are only 250 tons, a little more than 2/3 the size of the Webber Class fast response cutters. http://www.navyrecognition.com/index.php/news/defence-news/2018/june-2018-navy-naval-defense-news/6269-finnish-defence-forces-choose-kongsberg-naval-sonars-for-sq2000-mid-life-upgrade.html
Another light weight towed array. https://navaltoday.com/2018/07/04/new-kraitarray-sonar-ready-for-sea-trials/?uid=171
I know you’re thinking something small/light enough for Webbers, but wouldn’t the AN/SLQ-62 be the obvious choice, since it will be in inventory? Also, will something as small as the Webber class have all the necessary room for systems, weapons, and personnel to truly be active in a modern ASW environment?
Not familiar with the AN/SQL-62, and could not find any information. Sounds like an EW system.
At this point, we don’t know what sonar systems will fit on the NSCs or OPCs. I do think a good towed array system might be useful even in peacetime.
As for the Webber class, I don’t expect them to be ASW equipped unless there is a prolonged conflict. Still, they are more than twice as large as the Autonomous Continuous Trail Unmanned Vessel and it has both a hull mounted sonar and a towed array. Helos have also made pretty good ASW assets and they have small crews and far less volume than the Webber class. We would probably need a larger crew and a bit more crowding, but that will be acceptable in wartime.