Is the Liberty Dollar Network promoting an alternative currency to compete with Legal Tender?

No. Liberty Dollars are legal Silver Certificate Bearer Warehouse Receipts that are evidence of title to a specific amount of .999 Fine Silver stored with us, and may be freely traded with anyone. eLD is a system for the exchange of Digital Warehouse Receipts, but neither of them constitute "Legal Tender." Both are intended as mediums of barter, and either is redeemable in .999 fine Silver Bullion rounds or bars.