I realize there are many different instantiations of Construction Grammar (CxG), and I'm not necessarily tied to any particular version yet. I was curious if there are experiments that support CxG, or if evidence of serial (as opposed to hierarchical) processing is considered sufficient?
First, one big thing you're going to run into, almost no matter who you read: Most people in CxG focus their arguments against Chomskyan transformational grammar and other major competing approaches,1 not against non-hierarchical grammar or purely statistical approaches or other things that far outside the theoretical linguistics mainstream. For example, Simpler Syntax (mentioned below) does have some argument for why syntax can't be totally flat (and can't be parasitic off semantic structure), but it has a lot more argument for, e.g., why semantics also has to be hierarchical, or why syntax can't be as complex as current Chomskyan theory implies.
Anyway, I don't have a complete answer here, but… since the Parallel Architecture came up in the comments, and it also happens to be the CxG-style approach I'm most familiar with, I can give a partial answer, focusing on PA.
First, the "parallel" in the name is not about parallel vs. serial real-time processing (despite what the comments imply). It's about constructions being comprised of three parallel, independently "generative", structures: semantics, syntax, and phonology.2 PA was developed by Ray Jackendoff and Peter Culicover, starting with their 2005 book Simpler Syntax. While that's a pretty solid book, you might prefer to start with either Jackendoff's 2010 Meaning and the Lexicon, which gives a lot more historical overview, or Culicover's 2013 Grammar and Complexity, which is more up-to-date and more rigorous—and also much more firmly in the CxG camp.3
One of the intended goals of PA was to come up with a theoretical grammar framework that can actually connect up to psycholinguistics, unlike mainstream Chomskyan grammar. For example, Culicover argues that the hard competence/performance divide (going back to 1950s Chomsky) needs to go away, and shows how it can. All three books mentioned above contain references to a variety of experimental work, as do other papers and books within the PA framework by both authors.
Jackendoff has also, since the late 2000s, been collaborating directly with psycholinguists to gather experimental work that's specifically relevant to his theoretical framework. I don't know much about any of those papers, but you can see the Experimental Articles section of his recent publications page for pointers. Also, I believe Gina Kuperberg has done some independent psycho-/neurolinguistic work that's relevant, and probably some of the other co-authors have as well. Taking a quick look at The Processing and Representation of Light Verb Constructions, it looks like this is relevant work, if not exactly what you're looking for.
Another good starting place is Adele Goldberg, maybe the 2006 book Constructions at Work. She has her own CxG approach, different from PA. Like Jackendoff and Culicover, or probably even more so, one of her major focuses has been working out how to connect formal linguistics to theories of real-time processing, language acquisition, and language evolution. And her papers, like theirs, tend to contain references to experimental results. For example, her original paper on statistical preemption4 goes into detail about what kinds of testable predictions it makes.
One last point: Most of the arguments are pretty indirect. Let me give an example: Various pieces of evidence (e.g., timing tests and distraction tests) show that there's a higher cost to processing whatever-is-going-on with linking wh words to gaps when the gaps are in subjects rather than in objects (in English). And there's some evidence implies that topicalized objects act like subjects in this regard. That isn't prima facie evidence for CxG. However, under Culicover's CxG-based model of real-time processing, the explanation is obvious, if a bit involved (I won't go into it, but see, e.g., chapter 5 of Grammar and Complexity); there might be a similar explanation in a linear processing theory, or in transformational grammar, etc., but it's not obvious (at least not to someone who specializes in CxG…), and nobody's yet offered one. So, that's some sort of evidence for a CxG approach. But how strong? What if many of the alternative theorists haven't even read Culicover's work (which is entirely plausible), so they don't even know they've been challenged to explain these results? What about frameworks that only have a handful of people working on them, so they can't be fairly expected to answer every challenge they hear about?5
1. And, as another unfortunate side effect, they expect you to know some of the arcane Chomskyan terminology.
2. Most other approaches, both CxG and otherwise, have a single core generative structure (usually syntax, but sometimes semantics, as in some cognitive grammar approaches), and the others are assumed to be isomorphic to it, or parasitic off it in some other way.
3. Simpler Syntax is careful to try to avoid any prior theoretical commitment to CxG, and draws on predecessors ranging from HPSG to Chomskyan GB, even though ultimately the conclusions are pretty CxG-flavored. Subsequent work, by both authors, draws much more on SBCG, HPSG, and other CxG and closely-related frameworks.
4. I can't remember the name of the paper—but the idea is covered in the 2006 book, so you probably don't need it.
5. And that's effectively the case for almost every alternative except mainstream (Minimalist Program) transformational grammar.