I found a way, if someone is interested.
You need to download xaas-blueprint from vra (using cloudclient) and manually update yaml file. For "problematic" input change refreshOnChange to false. It looks like that vra automatically sets this to true for inputs that are used as parameters for data-binded actions (but there is no UI to change that back). This will cause that page will not be refreshed after input is updated, but that is exactly what we want.
Run parameter is still required since we will leave refreshOnChange = true and updating run parameter will trigger action.
Additionally, if you do not want to change your action with "run" parameter as input, than bind action with expression:
(#run : GetAction("module","action").call(#input) : " ")