Unfortunately, LAHD does require the new owner to register or exempt the units in the year that it is purchased. As Ron stated below, your best bet is to pay the fee and then appeal it.
The property, once it's registered, stays in the LAHD system. They continue to assess and collect registration fees until they are notified otherwise.
You may want to go back to 2006 records to see what how was registered then. Did you purchase the property vacant? Were there tenants occupying the units before you purchased it or was it owner occupied then? That may be a start.
Keep track of the dates and names of whom you speak to at LAHD. They can be difficult to deal with at times, however, you may find that "jewel" of a person who will advise you of the process.
Good luck and all the best,
Prudential California Realty