I was originally thinking of recommending you a wall plugged USB hub, however; it seems you already went that route and the hubs were unable to negotiate the power necessary for reliability. Because of this, I had to take a step back and think of an approach that would be guaranteed to run two HDD's at the same time. I ultimately decided to recommend you specialized hardware designed for your exact user case scenario.
In practice, this system acts as a hardware layer between your HDD's and you target machine with a mini-PSU strapped to the back. This system will most definitely be able to handle your user case scenarios. Here's a video of the enclosure in action. It will of course, need to plug into the wall with an AC to DC adapter.
- 2 Bay enclosure
- Internal SATA interface (I/II/III)
- USB 3.0
- 2 A Output Current
- 5 v Output Voltage
- Compatible with Windows, Mac, Chrome and Linux (2.4.x and up)
- Individual switches (inside enclosure)
- Supports RAID 0/1/JBOD and BIG
There is also a USB 3.1 variant for $86.99 (MSRP: 125.99) and a Thunderbolt 2 variant for $236.37 ($326.99). Personally, I would stick with the USB 3.0 variant I recommended above as SATA III only supports up to 6 Gb/s anyway. The other 2 variants are overkill unless you want fan speed control and/or configure the system for a RAID 0 (striping) SSD configuration and purchase SSDs actually capable of 750 MB/s+ combined simultaneous read and write speeds. I say SSDs because any HDD you buy is pretty much never going to actually hit the SATA III limit anyway.