I think the term "budget hawk" has been misapplied so much that it's essentially meaningless. Libertarians seem to want a Federal government as it maybe existed 100 years ago. Right wing Republicans aren't necessarily interested in lower spending or deficits, but claim "deficit spending" as a problem when the tax levels are higher than they like, or when the spending levels for programs they don't like are too high. Everything else is okay.

There are a few people who actually care about running a government that isn't running in the red for eternity, but that's not really what they campaign on or they wouldn't get elected, in my opinion.