There is nothing odd or screwy about this. 4U is not a subsidiary of LH in any way, shape or form. It is owned by the same holding company as LH. LH is a member of *A while 4U is not.
If 4U wished to join *A, I suppose that it likely could do so and simply piggyback the LH system, but it clearly doesn't want to.
But, bear in mind that international lounge access is relatively expensive and LCC's aren't set up around them. Lounge access isn't free, it's simply included in the ticket price and what OP wants, is going to cost more and that defeats the theory behind having a LCC in the group.