You know all about the coordinate system of the data. It is based on a system you know with modifications you also know. Most of the battle is already won.
Simply define your data in the coodinate sytem you know it has, even though you have to create the system yourself.
Create a new coordinate system definition, based on the known one, with the appropriate modifications (meter units, custom false origin), and give it a new, appropriate name. Asign that new definition to the data set.
You should then be able to project it into any other defined system you fancy, or project other data into that custom system.