It is difficult to read what your question is about. Your introduction is:
I really like the truncated normal but I want to make it discrete. I need a discrete distribution and I hate approximations (the best fit for my data is truncated normal but my data are discrete [0,25]) I am fixated on the kernel and the nice relations and ratios of the different values.
So you hate approximations... but what is the question?
I already answered why. Most people when they have such data use Normal or Truncated Normal at best.
This doesn't clarify at all why you want to discretize the normal distribution and what the problem is.
The problem is very general/broad and cannot be easily answered without the question having more focus.
There might be a reason why you 'really like the truncated normal'. But if you leave it open for others to guess why this is the case, then this information is not very useful.
- If you want some discretized truncated normal distribution, what are the properties of the truncated normal that should remain when the distribution is discretized?
- How do you want the discretization to be done? The underlying mechanism in your problem, why it is that a normal distribution is a good model, will influence the way to do the discretization.
If you explain the problem and the motivation then all such considerations do not need to be guessed.
And to be honest. I did not read the second half of the question. It is a big block of text which might become more readable if you split it into multiple paragraphs.