Hi Andy,
assuming you tried the aircraft.cfg tweaks to no avail. Sorry for trivia but Auto-throttle engagement would not work without the IAS hold (in some cockpits SPD) button selected.
Just as a test, what I usually do, actually take-off with Auto-throttle:
1/ Select AP runway heading
2/ Select the AP altitude 10000, VSI select lets say 1500 FPM
3/ Select speed dial to 250kts
4/ At this point all AP functions disabled and lined up for take-off, engines running
5/ Autothrottle ON
6/ IAS hold (or SPD) on - aircraft starts to accelerate
7/ Rotate command - rotate to selected VSI
8/ AP engage, AP heading engage, AP altitude engage
9/ Gear up
10/ Flaps up
At this point the speed stays at 250kts and aircraft climbs to the selected altitude maintaining the selected VSI. Then I change the AP heeding and altitude as directed by ATC. Increase AP speed dial after 10000ft.
This always works for me. Occasionally after take-off the speed exceeds 250kts slightly but reduces to the mandated 250.
P.