I'm not a licensed attorney and you should seek qualified legal advice.
That said, I've run a CPO firm under a 4.7 exemption. I've also had the same entity registered as a CTA, major swaps participant, and investment advisor (SEC) all at once. Needless to say, we traded all of the asset classes out of the same entity.
Likewise, you'd be able to trade equities (or spot FX, swaps, treasuries, anything really) as a CPO firm. Just that depending on the nature of your business, you may well be required to also file with state or federal regulators to register as an ERA, or file Form ADV with the SEC to register as an RIA.
There's obvious advantages to this approach, since the accounting, audit and administration work are consolidated and it becomes cheaper than operating entities separately. Moreover, multi-asset vehicles have appeal to large institutional investors. However the downside may be that certain investors have internal mandates that prevent them from investing in a particular asset class, and if your business is highly conditional on 1-2 of those as major investors, then you might prefer to have actual separate entities trading derivatives and securities rather than allocating the PnL and positions differently across different classes of securities offered by the "CPO" entity.