I would assume that this data would follow a normal distribution, so that is where I would start.

When the bin width is even, you can use the bin center as the x value and the bin height as the y. In your case, since the bins are uneven, you should use the bin integral of the objective function to compare to your data. For example the code below:

import scipy.optimize
import scipy.stats
import numpy as np
import matplotlib.pyplot as plt

bins = [0, 600, 700, 800, 900, 1000, 3000]
binc = [ 300, 650, 750, 850, 950, 2000]
weights = [340000, 365000, 494000, 430000, 110000, 40000]


def fGaussianCDF(bins, *params):
    
    N = params[0]
    mu = params[1]
    sigma = params[2]

    binwidth = np.diff(bins)

    return N*(scipy.stats.norm.cdf(bins[1:], mu, sigma) - scipy.stats.norm.cdf(bins[:-1], mu, sigma) )


fig, ax = plt.subplots(1, 1)
ax.plot(binc, weights, "ok")
ax.set_xlabel("Weight (lbs.)", fontsize=16)
ax.set_ylabel("Counts", fontsize=16)


popt, _ = scipy.optimize.curve_fit(fGaussianCDF, bins, weights, p0=[1.8e6, 730, 150])
plt.plot(binc, fGaussianCDF(bins, *popt), "rx")
print(popt)

plt.show()

Which gives the best fit result of a mean value of mu=736 lb and sigma=146. The results plotted look like:

Which is not a perfect fit, but hopefully is something that you are looking for.