If it were me, I'd go right ahead with planting - as long as the compost is not hot. With a marginal amount of dried materials left, I don't think you'll have any problems.

Since the bacteria that digest the material in a compost pile work best in an environment they like, and that environment is the center of your pile, the edges often are not worked on at all and that is the reason for turning your compost pile, periodically, to put the undigested stuff on the outside into the center of the pile so it will be worked on. Whether the undigested material will have an adverse affect on plants this fall will depend on the ratio of undigested material to digested material, because if the undigested stuff is greater the bacteria will work on that and not feeding the plants you want to grow. However, if there are several weeks between the time you filled that bed and planting that may not be a problem.

Is there something I could put on top of the half finished compost in the raised bed to complete the decomposition? Read somewhere that Coca-Cola would speed up the process. Plan to wait till October to plant cool weather vegetables.

You might try some blood meal and water it in, but I honestly think you're fine as-is.

The only thing that will "finish" composting the material on the edges of your pile is to turn them into the center where the bacteria will work on them.