A few items...
You obviously will not be able to address all of them with your existing P410 controller, but:
*Do you need all 16 drives in one array?*
I ask, because there isn't much utility in having that many disks in a single array unless you know your read/write patterns and are designing around it.
If you don't need all 16 disks in a single array, you're free to obtain a second Smart Array P410 RAID controller to address the second drive cage. That is the simplest solution.
There aren't any HP controllers that support the setup you're looking for without an expander. The Smart Array P812 controller is meant to accommodate internal drives, but is really intended to fan-out to external enclosures (D2600 and D2700, for instance).
All other options internally would require the use of the HP SAS expander. I use the expander in a few installations and have had good luck with it (once the firmware is current). If you have an old version or an old firmware, the expander only works at 3G. However, it can be a bit of a bottlneck, depending upon what you're doing (e.g. an array of SSD's).
The expander is a 6G SAS unit, so you're looking at a single 4-lane SAS SFF-8087 connection from your onboard Smart Array to the expander. There's an option to use a second connection... So that's either 24 or 48Gb/s... From the expander, you'll have 2 SFF-8087 cables going to each drive cage.
16 x 6G disks == 96Gb/s theoretical max. You'll be oversubscribed on your link to the controller by either 2x or 4x, depending on how you cable the chassis.
Let's assume your single disk is capable of real-life ~200 Megabytes/second transfers. Maybe slightly-lower: 1.5Gbps...
1.5Gbps x 16 = 24Gbps
That fits well with a single 4-lane 6 SAS connection. Even better if you use dual connections between the controller and the SAS expander.
So yeah, it's possible to stuff a DL380 G6/G7 with disks and get good performance with the expander. I think the separate P410 controller is a smoother solution, though.
The authoritative resource for the SAS expander is here.