Design of a differential digital signal buffer, 250MHz, 1V output swing
Thanks to the comments above, I realised my main problem was that I was trying to run with the current too low, and this was causing a long delay in the output, leading me to not notice that I had unintentionally inverted the outputs.

Increasing the bias current to 1mA made everything work a lot better (and made it clear that the hack with the inductor wasn't really necessary, as once the circuit was working better it didn't have any useful effects).

Indeed, it turns out the for a circuit of this nature having a stable current sink isn't actually all that helpful.  Which just shows that reading about analogue amplifier design when you're designing a digital circuit isn't necessarily all that useful.