To address the question in the title:
Probably any cross among those breeds will be fairly hardy.
The best layers will probably come from crossing the hens that lay the most eggs with the rooster of the best-laying breed (I would guess the Barred Rock rooster, but I don't know that for sure.)
An obvious, simple way to do it: let them all mix in any way they wish, then sell the chicks as "barnyard mix." I do not know how well those chicks would sell, but it is certainly one option.
If you want to control the breeding more carefully:
You might try doing some purebreds (Barred Plymouth Rocks, Silver Laced Wyandottes.)
Easter Eggers might also be a good choice.
It might be worth getting a test on the Lavender Easter Egger to see if he has one or two copies of the blue egg gene. If he has two copies of the blue egg gene, then you could cross him with all kinds of hens and be certain that every daughter will lay eggs in some shade of blue or green (because every chick will inherit the blue egg gene from him.)
If the Easter Egger rooster has just one copy of the blue egg gene, then he will give that to half of his daughters (so they lay blue or green eggs), but the other half of his daughters will lay brown eggs if their mother was a brown-egger, or they might inherit the blue egg gene from their mother if she lays blue or green eggs. If the Ameraucana-mix rooster inherited the blue egg gene from his Ameraucana father, he will also give it to half his daughters.
Testing for the blue egg gene:
https://iqbirdtesting.com/blueegg
Predictions of what color chicks from which pairings:
The Barred Rock hens can be used to produce sexlinked chicks with the Silver Laced Wyandotte rooster or the Lavender Easter Egger rooster.
The Lavender Easter Egger should produce solid black chicks with all hens except the Barred Rocks.
The Barred Rock rooster should produce black chicks with white barring from all hens.
The Ameraucana/Barred Rock rooster can produce black chicks (some with white barring, some without) from any of the hens. Depending on what color genes he inherited from his Ameraucana father, he might also produce some other colors of chicks from Rhode Island Red hens, Silver Laced Wyandotte hens, and maybe Ameraucana hens.
The Silver Laced Wyandotte rooster should produce black chicks from the Black Australorp hens, black sexlinks from Barred Rock hens (barred sons, not-barred daughters), chicks with silver & black patterns from the Rhode Island Red hens (these chicks will probably show some yellowish or red leakage as they grow up), of course he'll produce purebred Silver Laced Wyandotte chicks with hens of his own breed, and I can't easily predict what he'll produce with Ameraucana hens because I don't know what color they are.
