I recommend to change the focus of your retrospectives. Instead of finding ideas, try to focus on how to implement a single idea. Let's say, you have three parts of the meeting: finding ideas, what is the next step, and how to implement that step.
In the first part, you do the idea collecting - nothing new here -, but in the second part you pick only one idea, which the team will implement during the next Sprint. In the third part, you do a planning meeting like session on how to implement that idea and bring the result - the tasks - to your Scrum board. It is highly recommended to timebox these parts so that you have enough time for the third part.
With this approach you have your focus and tracking. There is no need to talk about the progress of the idea during on a daily basis, but it can help a lot. Putting the created "idea tasks" into the Sprint Backlog with a different color can also help, but this trick is team dependent.
If you don't have the improvements on your Scrum Board, they won't be visible, and this means that they don't exist (yes, you can have improvements on your Scrum board).
Budgeting is an interesting question. A good Product Owner knows, that continuous improvements are crucial in Scrum - the framework is about continuous improvement - so it is not a question whether to have them or not. If you still have trouble, then you need to do more coaching and help people understand the importance of continuous improvements on each level. Doing them off the record is not a good idea, because then there is no real transparency any more, which eventually will lead to mistrust between the Product Owner and the Scrum Team.