i have a very similar problem going on with my LS400
the tracOFF and CEL will come on while moving and when it dips down below 10-15mph it will shut itself off
last night it happened twice and it took 3-5 cranks (kept stalling) before it would stay on.
so you think it may be the alternator...
Well it just happened today, but after a cool 15 minute run at 95-110mph tailing a rabbit and pulling off at an exit ramp, the car (91 LS400 w/164k miles) stalls itself out, start it back up and cant keep it on unless constant gas is kept on it.
Pull it into a parking lot and let it shut itself...