It is not clear to me if it is more important for the OP to minimize the overall cost, or just that of the companion, and I infer that the OP is based in the USA.

If one is just looking for a reasonably-priced companion ticket, and is willing to buy a full-fare primary ticket, the American Express Platinum Card has a (not great, but OK) companion ticket program that gets you a "free" companion ticket when you buy a full-fare ticket in C or F.

LH is one of the airlines that participates in the program. By the time you pay fees and fuel surcharges, the ticket can cost over $1000, but it can be a reasonable option if the idea is a cheap companion at the expense of an full-fare primary ticket.

The card is also not cheap ($450 per year service fee), but if one is based in the USA, the card also provides for lounge access when traveling on AA, DL, and US (and US lounges are available even if traveling on another airline). There are also other benefits to the Amex card, and it isn't a super deal, but can save some money if one travels often on flexible premium-cabin tickets and is looking to take a companion along.