That looks like a high quality lens. Here are my thoughts and suggestions. I actually think the Z1 lens might work well as a precollimator lens. For high light collection efficiency you want to maximize the lens diameter to focal length ratio, but it’s hard to get this ratio much larger than 1 and still have good lens quality (from what I’ve seen). I think the Z1 lens has a ratio probably around 1 and I’ve heard it is good quality.
Here is how I would choose the location for the precollimator lens. Rig up the large lens so that it is in place, 16” from the LED. Then hold the Z1 lens and start very close to the LED. As you move the Z1 lens farther away from the LED, you should see the cone of light coming from the Z1 lens start to narrow. Keep moving it away until the light cone narrows so that it’s overfilling the 7” lens by 1 to 3”. Around this configuration is where you are going to maximize the light collection efficiency. This should approximately double the beam area and hopefully not drop the lux by more than 10%. Fine tune the 7” lens position to get focus because adding the Z1 lens will change this distance slightly.
See here for a picture and more discussion.