How do I configure a external RTC that uses 3-wire SPI for the LPC 2387?
You need to tie the MOSI and MISO lines together and follow the protocol in the product's datasheet which will ensure a half-duplex mode of your SPI peripheral. Note that the RST signal is high when you talk to the RTC so you need to invert your CS somehow.

MISO tied to MOSI might make it hard to talk to other devices on the same bus.

One way to avoid issues would be to bit-bang the interface. I suppose it's not possible to use another RTC?