You can have the check-out or payment button click added as an event on your sub-domain. All redirections coming from your sub-domain can be tracked on Google Analytics.
You can create a trigger in your Google Tag Manager (GTM) for all click events or assign an ID for a specific click element. Once the tag in GTM is fired, you’ll be able to track this event in Google Analytics
I am tracking the click button events, but those not necessarily turn into purchase. Users will need to complete the info at checkout and pay. I want to track the purchase conversion events, so I can calculate the conversion rate of my landing page.
After enabling eCommerce tracking, you can create a complete funnel. Also, when viewing conversions, you can use Source/Medium to see if the users are coming from there, to determine the impact on the conversions.