How to put a 74HC165 on an SPI bus?
can someone suggest some other parallel-in serial-out shift register
  that does have a tristate serial output line, and therefore suitable
  to use on an SPI bus?


Mouser, Newark, and a few other distributors and suppliers have a selection filter that helps when I'm looking for "something like a 74HC165, but with tristate serial output":

74HC589: 8-bit serial or parallel input; serial output; shift register with 3-state output. "The HC589A directly interfaces with the SPI serial data port on
CMOS MPUs and MCUs."

-- On Semiconductor datasheet;
Fairchild datasheet
(So it meets the "must have a second source" criteria that some projects have, as mentioned by Jason S).