2k seems high for a car service but probably not for a hotel. The airport is far out of town to the north and the LM is on the southern end of the city. It's maybe a 45 or 50 min ride w/o traffic.
If you're looking at being closer to Fort Kochi, I might recommend you consider the Brunton Boatyard or Le Colonial. Both treated us very well. The owner of LC also owns about a half dozen other hotels in the area. We didn't go in them, but I recall they were generally well reviewed.