Is it possible to use SPI on Redbot Mainboard?
From the silkscreen on the board you appear to be either mistaken or already re-purposing signals - pins 10 and 11 are labeled as used for Servo position command PWM, not for sensor input.

Therefore, as long as you don't use those signals for servos (or ignore the labeling and use them for something else) there should be no issue using pins 11, 12, and 13 in the ordinary manner for SPI, by finding them in any available place on the board, including the 6-pin ISP connector your identify.

If you did need to use pins 11 and 12 for another purpose you could still "bit-bang" "software SPI" on any other available set of four pins.

It is also vaguely possible that you could share functions by connecting bumper switches through resistors - when the software is driving those signals as SPI outputs, it will dominate the resistors, but at other times it can re-purpose the pins as inputs to read the bumper switch state. However this may require modification of Arduino library code to accomplish repeated mode switching.