Cheapest and easiest way to run a periodic Docker process
AWS offers fargate now, which is managed by them rather than being a service like ECS which you ran the servers for and they manage the control plane. Fargate is basically "serverless" ECS. You can use a cloudwatch scheduled even to run something occasionally on a schedule on a fargate ECS cluster. Therefore you're only paying for what you're using. Something to consider is fargate is quite a bit more expensive than ECS running on your own EC2 instance, but if you're running in frequently you'll likely see a pretty marginal savings.

Another option would be to automate something similar. So create an ECS cluster with an ASG backing it, and adjust scale to match ECS demand, and then use cloudwatch to launch the container. ECS can then scale up and down, but this can be tricky because this method is usually used to autoscale ECS clusters based on cloutwatch triggers, so you might have to hack around in cloudwatch and possibly hack together a lambda script, since you're technically not using it the way it's intended. In the end this would probably be the most cost effective, but in your case the cost savings could be negligible.

In my opinion Kubernetes is a ten ton hammer that you'd be using to hang a picture on the wall. It's way overkill.

You can explore those 2 options and see if they are cost effective, and easy enough to manage. Other container orchestration is usually going to be overkill, ECS has the advantage(usually a disadvantage) of being stupid simple and you pay very little and have basically no operational overhead for the control plane.