Why do your friends think you're nuts?! This is a great idea!!
I hope you can find a suitable tour with the above suggestions. I stayed in that area just a few years ago so I wanted to chime in with a few details.
I would say, the main bulb-growing area is not that huge, and since it's so flat and there are bike lanes, it's really manageable. I'm not sure you'd need to go from hotel to hotel at all, unless you want to visit some of the larger cities as well.
We stayed in Noordwijkerhout, in a bungalow at De Gouden Spar: http://www.goudenspar.nl/en/ The bungalows are busy in the summer, but in spring they may not fill up until after the hotels all do. There are several bungalow parks in the area, and there are hotels too if that's what you prefer.
Bungalows are self-catering, so you would have to find the nearby Albert Heijn and get your own groceries and cook your meals. De Gouden Spar even has a few spare bikes just laying around that you can borrow. Admittedly, getting groceries is way easier with a car, although you might be able to borrow or rent a cargo bike.
Noordwijkerhout is nestled amongst bulb fields, and also across the street from a vast network of sand dunes that are full of paths - excellent for exploring. The walking/running paths are separate from the bike paths, and you can find your way out to the beach.
Also, we passed by at least one bike rental place nearby, which I believe was this one: http://www.rentabikevandam.nl/index_en.html
There is a parade called the Bloemencorso that goes from Noordwijk to the Keukenhof, and from Noordwijk or Noordwijkerhout you'd be close enough to bike over and find a spot to watch from. Looks like it's Saturday, May 3rd at 9am this year.
If you can't find a suitable package tour, you might do well to rent a car and also rent a bike, and pick a hotel or bungalow that suits you. You can take the car to go to the Keukenhof early in the day, to get groceries or run to the drugstore, and/or to range around the smaller highways and get pancakes at a couple of the old-school pancake restaurants. You can take the bike to meander around the little towns and the fields everywhere, or to jaunt out to the beach if you get a nice day for that.