Vulkan imposes no requirement on the implementation that it permit direct usage of a swapchain image in a compute shader operation (FYI: "rendering" typically refers to a very specific operation; it's not something that happens in a compute shader). Therefore, it is entirely possible that the implementation will forbid you from using a swapchain image in a CS through various means.
If you cannot create a swapchain image in your preferred format, then your next best bet is to see if you can find a compatible format for an image view of the format you can use as a storage image. This however requires that the implementation support the KHR extension swapchain_mutable_format, and the creation flags for the swapchain must include VK_SWAPCHAIN_CREATE_MUTABLE_FORMAT_BIT_KHR as well as a VkImageFormatListCreateInfoKHR list of formats that you intend to create views for.
Also, given support, this would mean that your CS will have to swap the ordering of the data. And don't forget that, when you create the swapchain, you have to ask it if you can use its images as storage images (imageUsage). It may directly forbid this.