Please take a look to the following posts, which may be of help:
As mentioned in one of them, for simulations involving an Am-Be source, the recommended approach is to start the simulation directly from the neutron spectrum. To learn how to sample from a given spectrum take a look to
Then, seems that you are interested in dose equivalent (correct me if I am wrong), you can simply set an USRBIN scoring DOSE-EQ, whose results will be in pSv/primary. To transform to equivalent dose rate you just need to properly normalize using the activity of your sample.
Not sure what you mean by “I could not get the expected neutron dose rate”, could you elaborate?
Regarding the gamma, it is generated when you start from the alpha particle in the Be target but, since I just recommended to start from the neutron spectrum, you could run a second simulation, this time with a 4.43 MeV photon beam (instead of a neutron one). The dose can be obtained as in the previous case (USRBIN).