I'm really struggling to follow what you're trying to do - but why do you not just use the same writable store in both your page and template/component?
Also, maybe look into store subscriptions and/or reactive declarations for what you're trying to do.
Regardless, it sounds like you really need to just read the docs or do a tutorial.
On my page, I had a script tag pulling from the store. The variable was reactive within the script tag.
I export let that into the template via “data” props.
I changed the value of the variable within the script tag.
The template did not reflect the new value.