CNAME setup with multiple accounts

We are providing a service to our customer who is already using Cloudflare to protect their main domain in this example, we will call it customer.com. We are then wanting to manage an application which will use a domain name of app.customer.com as well as app2.customer.com but this is registered in our account. To do this we need to use the cname method and get them to add in the text record from the Cloudflare account which ends up with two text records for customer . com pointing to two separate Cloudflare accounts. The problem is that when we do this, the main site customer . com is going down. Is this a valid setup and if so, does anyone know why this might be failing?

Apologies for the formating, I am only allow 4 links for a new post which is difficult when talking about this issue

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