Did you try taking the wheel off and giving them a squeeze, that would make the pads self adjust further in and pull more fluid from the reservoir, so the pads will sit closer to the rotor and lever throw should be less, if you wanted you could probably put another drop off fluid in the system after.
I don't have any of the bleed kit handy so it'll have to wait but I was thinking of using a smaller spacer block and then adding more fluid with the method linked above.
I'm pretty sure it'll be fine with whatever is in the reservoir, at least until the pads wear down as there will be enough to compensate for that anyway. I'd just try what I said and avoid putting the bike upside-down.
Did you try taking the wheel off and giving them a squeeze, that would make the pads self adjust further in and pull more fluid from the reservoir, so the pads will sit closer to the rotor and lever throw should be less, if you wanted you could probably put another drop off fluid in the system after.