the ADC is LTC1859 and triggers on the slave select falling edge.
No, it doesn't. Here is a link to its datasheet. It has a pin labeled CONVST, for “conversion start”. According to the section Pin functions, “This active high signal starts a conversion on its rising edge.”
If you want your sampling to be at 2 kHz with cycle-accurate timings, you have to send to this pin a train of pulses which are at least 40 ns wide and have a period of 500 µs, i.e. 8,000 CPU cycles. This can be done with any of the three timers of the Uno. For example, you can set Timer 2 to repeatedly count from 0 to 124 with a prescaler of 64 (untested code):
// Configure Timer 2 for PWM on pin OC2B = digital 3.
OCR2A = 125 - 1; // period = 64 * 125 CPU cycles = 500 us
OCR2B = 2 - 1; // high for 64 * 2 CPU cycles = 8 us
TCCR2A = _BV(COM2B1) // non-inverting PWM on pin OC2B
| _BV(WGM20) // mode 7: fast PWM, TOP = OCR2A
| _BV(WGM21); // ditto
TCCR2B = _BV(WGM22) // ditto
| _BV(CS22); // clock at F_CPU/64
Note that the timer will also set a flag when the pulse is done, at which time you can start the data transfer. You would then start your data-taking loop with:
// Wait for the output compare flag.
loop_until_bit_is_set(TIFR2, OCF2B);
// Clear the flag.
TIFR2 |= _BV(OCF2B);
// Now transfer the data.
...
Edit: I would just want to add that cycle accurate timing does not mean perfectly uniform timing. The Arduino Uno is clocked off a ceramic resonator. This kind of resonator, in addition to having bad accuracy, has poor frequency stability. To get an idea of how bad it can be, see the Allan deviation plots on the article Arduino clock frequency accuracy, by Joris van Rantwijk.