Get random numbers from lists that follow Poisson Distr

Question

I have a simulation that produces each time a list of integers starting from 0 onwards

simulation 1: [0,1,2]
simulation 2: [0,1]
simulation 3: [0,1,2,3,4,5]
etc..

I want to choose one element each time so that

1. I can get a target mean (example 1.45)
2. The sampled numbers follow a poisson distribution.

I have tried generating those lists using a poisson distribution with a specific mean (example 9.5) and the meaning of those lists is how many times number 3 can fit as a whole inside. So sampling poisson(9.5) I may get

simulation 1: 8 -> the max number of 3s is 2, hence choose one of these:[0,1,2]
simulation 2: 3 -> the max number of 3s is 1, hence choose one of these:[0,1]
simulation 3: 16-> the max number of 3s is 5, hence choose one of these:[0,1,2,3,4,5]
...

Then my goal is to say my target 3s mean is 1.45 for example and

1. from simulation 1 chose 0
2. from simulation 2 chose 2
3. from simulation 3 chose 1
etc...

What I tried: Split each simulation in N-1 spaces using the Poisson(1.45) CDF

prob 0: (0, 0.2345)
prob 1: (0.2345, 0.5745)
....
prob n: (poisson(n-1, 1.45), 1)

and choose based on those probs, but my method fails. I do not get the expected mean, and even if I do, my distribution is not poisson.