OP, do you realize that you will be entitled to use Sky Priority check in, which also has its own very separate area? At LAX there are 3 main check in areas:
Delta One: for (most) international & transcon Delta One customers
Sky Priority: For domestic first class & medallion members (excluding silver medallion)
General check in: all other members (includes special lines for silver, group travel, etc)
When you are being dropped off at the curb, look for "DELTA PRIORITY"