Could use the Beaglebone Black, it's has more than enough GPIO pins and has dedicated coprocessors for bitbanging, which improves throughput and latency.
Think with the BBB you have to be super careful with those GPIO pins (more so than the RPi apparently) - have heard of people having fried their Beagle as a result of a tiny mistake.
The GPIO pins on the BBB are 3.3V tolerant, but not 5V tolerant like the Arduino. RPis GPIO has the same power constraints as the BBB (3.3V tolerant, but not 5V tolerant).
It's fairly straightforward to check if you're going to need a level shifter, takes seconds to check voltage with a multimeter.