I have been a sourcing agent in China serving amazon sellers for years, I'd like to give you some personal opinion in my perspective.
yeah, $1000 budget maybe a little tight, but dose not mean impossible, I know you want to control the risk, but it should not be a rigid restrict.
you'd better start with 1 product first, cause newbie sellers usually have to learn a lot in the first product, like amazon background operation, suppliers management, amazon policy, etc. Once the stage is suitable, you can expand your business by trying other products you researched.
you can even have a try to learn sourcing some products from China to lower your cost. This blog may help, How to Import Small Quantity from China: Guide for Business Starter
Hope your amazon private label FBA business go booming.