How to create a band limited signal of sine waves that is nearly uniform distributed
This is a wierd thing you are trying to do. What i the purpose behind this signal generation? What particular application needs this?

I think of two ways, but maybe one is not correct.

1 - Generate Uniform White noise, and then filter it to your desired bandwith of interest. But that will surely affect the probability distribution of the amplitudes and make it more laplacian, like your graph shows.

2 - Generate Any signal you want with the desired narrow bandwith (Still not sure why you need this, but let's say you do). you could use method one for it. Then estimate the probabiliy density with any method, like periodogram like you did, or Kernel Density estimation with parzen windows, etc.
Then, design a non-linear transformation that you can apply sample to sample, in order to go from the probability density that you have, to a uniform. This non linear transofrmation is actually the "Inverse Probability distribution" that you have.

If you estimate your density in discrete intervals with periodogram, maybe you could look at the form of the graph and try to guess what functional form it has, then fit it with least squares to a continuous function, and then obtain the Probability Distribution from that, find the inverse, and apply that transformation to every sample. That should give you a signal which is uniformly distributed. 

I am guessing now, but i think that your bandwith will not be preserved by this transformation. I have actually never studied the relationship between bandwith and probability density of the amplitudes. But try it and see what happens.

hope it helps.

Edit:
I think you are trying to do is Least Squares fitting. It is the most mathematically correct application of "averaging to get rid of interferences and unwanted noise", or of linearizing a non-linear systems. Take all your measurments, and fit them to a line. There are 5 types of Least Squares methods:


Least Squares with $L_{2}$-norm
Least Squares with $L_{1}$-norm
Least Squares with $L_{\infty}$-norm
Total Least Squares
Data Least Squares


Try some of them and see which one gets better result for your particular application.