This is a really great use case! So sorry for the delay in getting back to you!! Unfortunately, there is not a great way to accomplish this using the public RStudio Connect Server API today. However, it is possible to accomplish the first item in your query using an experimental R package called connectapi.
The parameters themselves are unfortunately not stored in an accessible location, so there is no real path to retrieve the parameters today 
But I personally am very excited about these types of use cases and am looking forward to sharing this with the product team!!
library(connectapi)
client <- connect()
my_content <- content_item(client, "my-item-guid")
get_variants(my_content)
myvar <- get_variant(my_content, "variant-key")
myvar$get_url()
I hope that helps!! Please let us know if you have any feedback on this workflow and what other components of your workflow are challenging that we could help with inside of Connect!