In short--yes--a gennie of that size would do what you want it to do.
In my experience, Honda generators are highly valued, especially by preppers because they run relatively quiet. However, I don't think you can get a Honda in your price range.
You might want to put this question out there to the board in general ... I'm sure you could get a wide range of input from people with experience with different brands and from those with professional-level electrical knowledge.