While I’ve used bus interfaces on microcontrollers before, I’ve never had to push the amount of data that you are looking to do so I’m not well versed in the max performance you can achieve with commodity development boards and lower end microcontrollers.
The most I can add is to take a look at the Zynq based boards. Zynq is Xilinx product that consists of an ARM core surrounded by FPGA fabric that can easily support your data transfer needs. If you don’t have a history with logic design using HDLs, this may be a bit of a reach for you since you will have to design the bus interface yourself unless you can find some usable IP that already solves the problem.
Some possible boards are the PicoZed and MicroZed from AVnet which can be found here: http://zedboard.org/
There a free (if downloaded electronically) book that goes into great detail about the Zynq infrastructure and it includes practical examples on readily available demo boards. http://www.zynqbook.com/
You’re in a tough spot with your requirements. Higher end microcontrollers with EBIs can meet your 10MB/sec acquisition rate but will be hard pressed to efficiently interface with the amount of memory you require. As you transition into the world of microprocessors which have the bandwidth and resources to work with memory other than SRAM, you start being burdened by the potential need for an OS of some sort.
What exactly are you trying to do and are you sure there isn’t a more efficient way of handling your data instead of buffering your entire capture and writing it afterward?
I know you don’t want to do logic design but a sub $100 FPGA board could acquire your data and simultaneously write it to an SD card without having a huge amount of onboard memory.
I can’t suggest certain SBC because we have different preference so, I found SBC comparison article in sparkfun. I hope this article will help you somehow to select the right SBC for you.