Thanks for doing that, Annika.
I have just checked for this Carl Eric in Arkiv Digital. He used the surname Mellberg and died in Kila parish in 1924. So not him.
ArkivDigital is the absolute best, and has you say, it costs. They do have short-term subscriptions and the occasional free weekend (which I always forget to look out for).
I suspect that a short-term subscription is not enough to learn the workings of the site and the indexes with vague data like this case; it's different if you have an exact birth date and/or a birth parish.
I did a search yesterday and found one Carl Eric with Gustaf Carlsson and Karin Andersdotter parents (no military connection, that info came later). Since, in the end, this guy also lived out his life in Sweden, I didn't find it worth posting. The names are really very common.