I agree with you that this is not a very convenient way to do something! The question you ask is your real goal to plan hours of service or to ensure the optimal work of your team, working both with user histories and with defects while using high-quality code, including bug fixes?
I would take it one step further from what Derek suggested - and use Kanban AND Scrum together - Scrumban is groping more and more! Since you said that you can have from 0 to 5 defects in any sprint, it is obvious that your “demand for failure” is variable, and therefore it is necessary for your “process engineers”. What do they do when there are 0 or 1 or 2 defects? I believe that they also contribute to the “demand for value” - new user stories.
This is where Kanban shines. While the actual design of your Kanban board needs to be reviewed by your team, you can start with a simple 2 swimming panel that reflects your current process to complete your work. Below is a simple example -

Here you have all your engineers to work in both lanes. As the work proceeds, depending on who can accept it and MAY accept it, they work on it and work on it. You are still doing sprint stuff in Staging - and deploy the package in one go.
In addition, you can have completely separate bands for user stories and defects -

Here again, all your engineers are working on objects in both lanes. However, you have the flexibility to deploy defect fixes as soon as they are fixed and accepted by the client (if applicable). Given the need for value, you continue to perform the same process as you do now, and deploy when each sprint is completed.
The benefits of any of these approaches are
- You get more people to work in any situation.
- You potentially get faster response time, better SLA performance, for defects.
- You get a happier team where everyone can work on new material. Most engineers do not want to be the “maintenance” guys :-)
Of course, this is based on a basic analysis. If you are not familiar with Kanban or Skrumban, you should read the books of David Anderson (Kanban) and the documents of Kori Ladas (Skrumban) and some others, such as Yuval Yere, Jim Benson, Masa Maeda and prepare for the best. You can also contact us at www.swiftkanban.com and we can also help.
Hope this helps!