After a bit of struggling and poking my nose through the happex board, I figured out that there was an offset of ~17.5us before the timing board accepted a trigger.
The logic to get it to work is a bit ugly but it works:
Split a TTL signal from a function generator at 960 Hz with a pulse width of about ~100 us (the smallest it can produce at this frequency). One copy goes to a gate generator that will produce a pulse of ~20 us. The other copy gets inverted by a flip-flop.
Take the output of the gate generator and the inverted function generator signal and combine them with a linear fan out. The result is a very weird looking pulse, but it has all the right properties we need.