How to use app registration on Microsoft online exchange

We have 2 tenants:

1 where we have credits granted to our Startup and registered under "PII ******@gmail.com".

A second one registered with the domain "caremcare.ai" which is the real domain of the startup.

We need to create an app registration on the subscription where we have credit (tenant PII ******@gmail.com) but we want to use this app registration to send emails "on behalf of" user belonging to the caremcare.ai domain. The objective is to send automatic emails using one of the accounts we have under the caremcare.ai domain.
The problem is that under Microsoft online exchange we do not see app registration belonging to other domains. How can we solve this problem? Is there the possibility to migrate the subscription with all the credits the correct domain of the company (caremcare.ai) instead of using the current one (PII ******@gmail.com)?

BR,
Salvatore Valastro