Sum of Discrete Uniforms, but each value can be picked no more than N times? Suppose there are i.i.d. variables $X_{1,..n}$ with discrete uniform distribution with the support $[1, n]$. What would be the distribution of such a sum if we introduce the condition that each value cannot appear more than $N$ times in the sum?
E.g., if $N = 1$, we have sampling without replacement, if $N = 2$, any number can only appear twice in the sum. $N = \infty$ is equivalent to sampling with replacement.

Let $n = 20, N = 10.$ then the population from which we can sample (without replacement)
consists of $N$ copies of $1, 2, \dots, n.$ Then the maximum sample size is $200$ and a sample x of size $200$ will exactly match the population. Using R:
pop = rep(1:20, N)
summary(pop);  length(pop);  sd(pop)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   1.00    5.75   10.50   10.50   15.25   20.00 
[1] 200       # population size
[1] 5.780751  # population SD (with denom 199)

x = sample(pop, 200)
summary(x);  length(x);  sd(x)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   1.00    5.75   10.50   10.50   15.25   20.00 
[1] 200       # sample size
[1] 5.780751  # sample SD

That part is trivial. So your question must have something to
do with samples of size less than 200.  However, you give no
clue what that question is.
Let's look at a sample of size $50.$
set.seed(2021)
y = sample(pop, 50)
summary(y);  length(y);  sd(y)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   1.00    7.00   11.50   10.92   15.00   20.00 
[1] 50
[1] 5.382663
table(y)
y
  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 
  2  2  2  1  1  3  3  5  2  2  2  4  3  4  3  3  2  1  2  3 

