Working on it although I don't know if I'll get it working. I've also searched far and wide for a library which might implement this functionality and it's not a thing. Will hopefully remember to post a solution if/when I come up with one.
Do you have access to an ArcGIS Server? If so you could create a model that uses tools from the Linear Referencing toolbox and then publish it as a Geoprocessing Service.
If ArcGIS Server is not an option, and you need to do everything client-side, you might want to look into Turf. I haven't had a chance to try this library yet, but the along and lineSlice functions might work to get you a line segment.
I think, Linear Referencing toolbox would only work if you have Esri Roads and Highways desktop and server extension installed. Check Measure to Geometry (Operation) , this will give geometry of the measure you provide. Only thing, you need Roads and Highways extension.
You've hit the nail on the head Jeff. It's a completely offline environment so everything needs to be done client side. I'm looking through Turf now and it is promising
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