If you have already created you linear referencing system. One way to approach the problem, how we do it in house, is to programmaticly query your LRS for the route referenced by your point, with the route filtered to the route in question snap the point to the route compute its measure (Distance along the route) and its offset and direction offset.
If programming is not your forte, (I don't believe python with arcpy can do this, need to use .net), you may want to try using the tool ArcToolbox/Linear Referencing Tools/Locate Features Along Routes this will give you the location along the closets route to the point (it does not use any filtering as described above).
Hope this gives you some ideas ...
After some addition thought ... you could combine within a model ( Read point route reference --->Filter LRS for that Route (memory layer/dataset)---> Run the ArcToolbox/Linear Referencing Tool.... with the inputs being the memory layer/dataset) Models are not my strong point.....