Kaihl choc switches (the V1 type) are going to have the shortest travel out of the keyboard switches that are common enough to use, and that’s what most low-profile hobbyist mechanical boards out there use. There’s a “choc amber” version that recently became available that have the shortest travel distance among choc switches (1mm actuation, 3mm overall), but they’re pricey and I haven’t seen anyone fill a keyboard with them yet. Those are also designed for fight pads, so I don’t think they’re made with silence in mind.
There are “silent” choc switches on their way (developed by Darryl lowprokb.ca), but they’re not available quite yet.
There’s also the Kaihl X switch with a 2mm overall travel distance, but they’re also pricey and I think it next to impossible to get key caps for them (at least it was last time I checked). Hazel is the only person I’ve seen make a keyboard for those: github.com/hazels-garage/shortstack. Awesome stuff, but probably not easily obtainable.