But. Come to understand the meaning of a sub panel and the implications for the neutral and ground.
You only want the neutral and ground bonded in one place in the whole system, and that is at the breaker that is most immediately downstream of the meter, that is your 200 amp disconnect.
At the original panel the neutrals and the grounds need to be perfectly segregated, and the neutrals need to be isolated from the case of the panel and the ground needs to be bonded to the case of the panel.
What sort of wire is the house wired with? If it is conduit, then the question of the ground is resolved, you need only find a way to isolate your neutral buss bar from the panel case. Sometimes this is easy, sometimes not. It could be easiest to just mount a new bar with the necessary insulation.
If your house is wired with NM or MC, then you need to direct all the neutrals to an isolated buss bar and all the grounds to a bonded buss bar. And have the #4 come to this bar, with a large lug if needed to make the connection.
Are there any large current 240v circuits in the house, like for the range, stove, oven, water heater, ac, etc? Until a few years back it was accepted to mix the ground and the neutral on these, this would be a three wire arrangement. Now we need to provide two hots, a neutral and a ground. The conduit can be the ground, but if you are going to go to this much trouble, do look at this and consider making it modern. If it is in NM, it will be more difficult, you would need to replace it absolutely.
That is a small fraction of what you are going to be doing.