Interesting. For starters, I think you have relevant experience and a good mindset: trying to find a way to communicate with people is critical, and often overlooked by most (at least, in my area), but as Tom pointed out, you're missing hands-on experience with actual development of product, in this case, games. I can't speak for other industries in software, but my impression is that games, moreso than any other related industry, require hands-on experience more than general qualifications.
Case in point: I've seen a lot more QA, programmers, designers, etc. promoted to Video Game Producer (or assistant producer) than I've seen successful producers coming in from other industries. And of the few that I did see come in, almost all were eventually let go.
That being said, I have yet to see a humble aspiring producer that has your instincts.
I know some (such as Tom) may disagree with my suggestion, but I believe it remains a valid one, and that it worked for many people, including myself. I shipped myself to this industry with no hands-on experience and started as a QA (you can see some of that possibly still pinned in these boards) roughly 6 years ago. I believe QA is a right fit, if perhaps quite a hit from a hierarchical standpoint, simply because it forces you to look at the end-product (games) under a much different light. If you're curious enough, it leads to a lot of learning and some great opportunities (I was a coordinator after roughly a year).
I'm not suggesting you take the path of QA, but possible, apply for a position that is not directly tied with running the project. A QA director/lead position, or translation assistant project management position could be a great fit as it would put you in direct contact with the project and the learning curve would lessen if only by osmosis with the environment.
If you intend on going head-first into production though, here is how I'd recommend going about it:
- You need design experience? Try to describe a game you know in over 1 and under 5 pages by breaking down the mechanics. Flowcharts may be relevant, but you can forego wireframing, etc. Stick to the essential. This won't make you a game designer, but it will help you struggle with some of their challenges (and remember, that's based on a game that exists!) All the software you need: Microsoft Word, or Open Office.
- Scrum Certification? In my own experience, a lot of employers appreciate the fact they can send some of their employees to these, as it gives them some form of PR, but they rarely hire based on it. More importantly, the people they send, they know they don't need to be afraid they could get "brainwashed". Don't get me wrong, there's nothing wrong with Scrum, but it is very rarely applied "as is" (and much less in servicing-based studios).
- Building your own game is a plus, but perhaps more on a personal level. I don't think you can easily "cash in" that kind of experience in an interview, especially if all the examples you have to talk about come from a single game project. That being said, once you DO get a job, it will make things easier for sure. I've used Unity for a bit now, and it's an engine that has gained a lot of popularity, so on most of the projects I work on, I tend to understand very quickly what we're talking about and it saves hours in meetings! (that's profitability/value right there that upper management can't deny).
A bit more on the interview (anyone feel free to jump in).
I think what you'll need to do is find creative ways in which your current experience is relevant to the role. This won't be easy, but focusing on resource management, conflict resolution, timeline management, etc. could help a lot.
Is there overtime in your field? If so, is this passed down from upper management, or do you have a say in this? Do you call for overtime or try to avoid it? Are you able to lift your team above expectations? How? etc.
If you can keep the discussion flowing around what you CAN do and are honest about what you have to learn, then there might be a business in a precarious situation that may be willing to take a risk with you.
Any studios local to you? Willing to move?