If you are reading about Toyota Kata, then you probably also stumbled upon Kaizen. Kaizen is about "changes for the better", even tiny improvements (but with time add up an start making a big difference). You can also use it as an effective personal development method. One big change might seem scary, the goal intimidating perhaps, but making very small steps in the direction of your goal is not as hard or daunting. So, with Kaizen, you can ask yourself, "what's the smallest step I can take or the smallest thing I can do right now to improve on what I'm doing?".
You ask yourself this every day, or every time you find some problem, or inefficiency, or waste you want to eliminate. You think on something to change, you act, you observe the result. If it's working keep doing it, if it was a mistake, correct it. With time, it adds up. This is in tune with what Agility means:
- Find out where you are
- Take a small step towards your goal
- Adjust your understanding based on what you learned
- Repeat
From your question you seem to consider such a process a large effort that takes up time from customer value creating work. Sure, you can have workshops, meetings, larger goals that you might need to spend some more time on, but it doesn't need to be like that. Small daily reflections and tiny changes don't register much on a sprint duration for example, but with time they can create a lot of positive change. And the total time they add up to the work, you often take back by doing your work more efficiently and being more effective.
In Scrum for example, you have a Sprint retrospective where you reflect on your work and try to introduce improvements. This isn't only about the work itself. This is mostly about the way you perform that work. Scrum prescribes this meeting to tell people that they should pause to reflect on what they are doing, or otherwise people might be focused so much on the work that they might neglect to find ways to improve it. For example, it takes discipline to do Kaizen. To do the work but also think about the work while you are doing it.
Read more into kaizen and how to apply it in your development process. And remember that you need some slack in your process to be able to inspect, reflect, and adapt. If you fill all your time with work, you will do the work in the same way forever, which might be OK if you are lucky, but maybe not.
EDIT after question being changed:
Edit to clarify some misunderstandings: I'm not talking about small problems we encounter and fix along the way. Those we're pretty good at. What I'm thinking about here is setting up a target condition towards a challenge and then deliberately and actively taking time to experiment toward that target condition.
As I mention in my comment, you need to look at your process from a different angle. Right now you don't have room for working on various goals because you are prioritizing the work for customers and are trying to optimize for resource optimization (which is a myth) when you should be trying to optimize for outcomes.
If you want to work towards some target to improve you process, then you need to take the time. Process improvement is important for a whole bunch of reasons, it's not subordinate to work you do for customers. It's then a matter of planning this effort just like you plan all your other work.