No, you won't need 2000, but you may need additional juice as a "buffer" some things that are powered require additional wattage to "start up". if you had a 1K generator powering a 1K light, would probably trip the breaker.
Go 2K and you have some versatility. That size will also come in handy during a hurricane as well. I have a 2900 watt and can run 2 fridges, a chest freezer and a 500 bottle wine cabinet.
The Honda EU2000i is probably the best small generator out there
. Small, quiet, and you can power your laptop off it without problems. I have powered 2, 1000 watt lights off mine, but it sure didn't like it. Try to allow yourself at least a 25% buffer. (2KW genset, 1500 watt load)
I run a EU2000 watt generator to power my 1000 watt stage light. With the eco switch on the generator only runs as much rpm needed to power the light which is a little over half throttle and it makes it really quiet. I looked around online and found one on sale for like $750 with no tax and free shipping good luck.