Oh this is a long question to answer, but let me start with my experience. I'm the "outsourced COO" for a number of small businesses who have grown through their bootstrapping phase, proven they have a market and all of sudden start feeling the pressure of the business expanding. 9 times out of 10 what they need is someone to wrap their arms around all of the operational stuff, mature their processes, put systems in place, connect their data, introduce greater discipline in the production of their core offering, make sure they have the right service providers in place (lawyers, accountants, bookkeepers, etc.) and they can easily and quickly get access to the right metrics to understand how the moves they make affect their desired outcomes. This is an important and often ignored role and is the key to relieving a hard-working CEO of the things they likely don't do "well", and allow them to focus on building their business, usually on business development and strategy. I think it is smart for you to consider adding this to your roster, particularly if your time is better spent on more strategic work.
More and more I would say that sourcing this kind of talent from recruiters and executive head hunters is becoming ineffective. You are best to scour LinkedIn and use your personal network for recommendations. You are far more likely to find good people this route than relying on the "you get three options" approach that many agencies will offer. You know your business best and who is going to fit.
It sounds like you are still relatively early stages. Don't worry about titles. The right person will be someone who has been there before, in a growth phase of a company, and knows that titles mean nothing, other than to better define their scope of work and help the employees understand their role. If you find the right person, they will be able to tell you exactly what you need from an operational perspective. I fear if you try to turn this into a CEO role, you will find someone who doesn't want to get mired in the details and focuses on the strategic work (appropriately), so you will end up with the same problem. Go find someone who focuses on, and is excited by building the operational foundation for your company. Someone who is going to say "OK, go build your business. I will make sure everything works to support that." They should also know how to scale operations to meet projected demand so you can work as a well-oiled machine.
Additionally, look for someone who has run a company before. I certainly earned my stripes when I went through this myself. Your candidate should understand business process engineering, change management, resource management, service design, financial and data management, systems and technology, etc,
Don't skimp on compensation. Pay them what they are worth and make the offer attractive. You will lose senior resources in a heartbeat if they don't believe you value their skills and are ready to make them an integrated part of the team, and to trust in their experience. Senior resources are not motivated by money but by the challenge, the team, the knowledge that they can be successful.
I hope this helps. Happy to chat further if you have some more specific questions.