Precision 15 keel, or capri 14 keel should keep you under the $2k limit... but hard to find...
"CaptnFoolHardy" sorry, forgot the Precisions, also great little boats. +1
You should be able to pick up a 20+ yo capri 14.2 centerboard for well less than $2k with a trailer, but be prepared to work for it, but again, reminder the centerboard capri is a tough learner. Precisions seem to hold their value more, and are harder to find... I have no experience looking for a lido.
An Oday 15 DSII I think, is also a decent boat, little easier than a P15, or a Capri 14 centerboard, but still a lively/challenging first boat.
Unweighted centerboard = potential capsize (generally), meaning expect to learn to get back into the boat. The smaller boats (laser, sunfish, force 5, etc) are easier to right, and generally easier to get back into. That being said, they are generally a wetter ride, too.
Hence if you want bigger, try to find a keel, it wont' guarantee no capsize/broach, but it will make it a lot harder to do.