Probably - you would still be drawing 6-7A - actually at the instant you turn on the key, you will be pulling about 10-11A while the pump is running (for about 3 secs after key-on).
If you like, you can repeat that extra load by leaving key on and just turning the Kill/Run switch off and on - lights will remain on regardless of the Kill/Run switch position.
See what the voltage is and how rapidly it is dropping as you leave it on.
it should be at least in lhigh 11/s to low 12's even with the lights on (and I would say the latter since you have LEDs)
And it should NOT be dropping rapidly.
I have LED lights and they barely blink when I hit the start button.
This one is potentially a big clue .....
If you hold the starter button (with clutch pulled of course) the lights should go completely OUT (even if the starter does not run) !!
(if outside and it's still light out, you can put the hi/lo switch on Hi and just watch the high beam idiot light - which should go off along with the lights)
If they do NOT go out, then the headlight/starter relay is not energizing
That could be
a) Relay itself
b) Start Switch
c) Clutch switch.