Not to keep beating the same drum ...

I suggest you look at something like the 23A640. You can do sequential reads and sequential writes. And all this over SPI. At 10MHz (and four lines - max), a 32-bit transfer is 3.2uS (that should be pretty fast).

From datasheet:
If operating in Sequential mode, the data stored in the memory at the next address can be read sequentially by continuing to provide clock pulses. The internal Address Pointer is automatically incremented to the next higher address after each byte of data is shifted out. When the highest address is reached (1FFFh), the address counter rolls over to address 0000h, allowing the read cycle to be continued indefinitely.
If operating in Sequential mode, after the initial data byte is shifted in, additional bytes can be clocked into the device. The internal Address Pointer is automatically incremented. When the Address Pointer reaches the highest address (1FFFh), the address counter rolls over to (0000h). This allows the operation to continue indefinitely, however, previous data will be overwritten.
This, or something similar, should fit your needs.