Short of having an existing pair of lavender mottled OEGB's...Or is there anything that will increase my chances (of course other than hatching 32+ chicks at a time

) I need a much larger coop and run Stat!
Coop size: yes, you may need a larger one, but early culling can really help. (Culling in this case includes selling or giving away, as well as killing.) The F1s will all look a lot like each other, and of course you'll keep some of each gender.
But when you start hatching F2 chicks, you will be able to recognize the lavenders at hatch, and probably also the mottled ones. I'm not sure whether you will be able to tell lavender mottled apart from plain lavender, but if you can that's even better.
Any chick that does not show lavender or mottling can be culled right away. This will mean you're only raising about half as many chicks, and the ones you are raising are the best choices for further breeding.
Of the lavender F2s, about 1/4 should actually be mottled, 2/4 should carry mottling, and 1/4 do not have the mottling gene. Likewise with the mottled ones: 1/4 lavender, 2/4 carry lavender, 1/4 no lavender gene. So crossing the F1 male to the F2 pullets that are lavender OR mottled gives an improved chance of getting chicks that are lavender and mottled. And again, you can cull the black ones really early.
For hatching lots of eggs: two incubators can work really well. Every week, you add another batch of eggs to one incubator. Every week, you move the eggs that are due to hatch into the other incubator, where they hatch. Then you clean the hatching incubator in time for next week's batch of eggs.
If you're hatching chicks every single week, especially if you then cull half or more of them, you can combine some of the age groups so there are fewer brooders/pens to tend. It's usually safe to put day-old chicks in with week-old chicks, and it's probably OK to have a few two-week chicks in there as well. You would generally want the little chicks to outnumber the big chicks, but that is exactly what happens if you cull a few more every time you can recognize that they aren't what you want. Some cockerels can be identified as early as 2-3 weeks, so you might cull all of them that are not lavender + mottled. (Because crossing F2 pullets back to the F1 male is about the same amount useful for breeding purposes, and helps keep down the number of birds you are dealing with.)
If you don't have an easy way to give away or sell chicks, you might be disposing of many baby chicks, or you might raise them for a few weeks and then butcher them like quail (tiny birds taste fine, they're just fiddly to deal with.) But there is no reason to raise them to full maturity if you know they are not what you want for breeding.