Luckily Bernadette and Kym at the Mall of America IKEA were there to suggest and secure the parts for me to make that happen. The Kivik one seat section could be modified to attach an arm from the loveseat and easily fit the space. Kym knew there was a set of these arms in as-is, and called to ask them to sell me just one.
I bought that and a new one seat section and took it home that day, but the cover was a tricky issue. IKEA doesn't sell just an arm cover, and the cheapest set including an arm in the fabric I wanted (Tranas black) is $189. After thinking about it over the week I decided to go instead with Ingebo red for the seat and Blekinge white for the arm. The Blekinge white sofa cover set cost $49, and the extra cushion covers may come in handy as a spare set.
The total expense to update my living room including the sofa, modified seat, covers, and a $20 Lack coffee table came to under $800. This is what the one seat with one arm looks like finished: