As you've noted, your tanks weighs more at the beginning of the dive than at the end, so since you are okay with your weights at the end, then you should have no problem at the beginning.
If you are using a vest BC, then try this: Float vertically, hold the BC deflator up high and press until all the air is out (some BC's only empty completely when you are vertical). Simultaneously you should exhale all the air out of your lungs, and either use you hand to push yourself under water while doing this, or kick yourself up so as to then fall down below the surface which will give you the initial momemtum to descend.