High frequency SPI-like communication: is there any hardware allowing software tweaking?
SPI peripheral inside a general purpose microcontroller (AVR, or PIC, or MSP40, etc) is hardwired to a large extent.  It is what it is.  Transaction stop sequence is formed by the CS# line.

Want to use a non-standard SPI-esque communication format with framing bits?  I think, it would be safe to assume that there isn't a general purpose microcontroller (μC) with a flexible enough SPI peripheral that can support that.

Probably, a good bet would be to use programmable logic (similar to your idea about external shift register).  You could program a CPLD (or FPGA) to convert from non-standard SPI-esque protocol to normal SPI, then a general purpose μC would read the data through normal SPI.  There are also Cypress PSoC microcontrollers with built-in CPLD fabric.