Hello @Xristoforos Papas !
Welcome to Microsoft QnA!
As i understand there is a System ( collection of services ) running already on an Azure Subscription and you want to move it to a Customer Tenant.
You need Access on Both Directories for the Move to complete
For the Move Operation of resources it is supported and here are some details
https://learn.microsoft.com/en-us/azure/azure-resource-manager/management/move-resources-overview
BUT
My approach is already on your mind
Transfer the Subscription into the Client Tenant
It is a common practice , it is safe and you wont have to make System changes
Of course if anything is connected to Azure AD has to be carefully planned for the new Azure AD , as Identities and Role Based Access Controls
Of course if the System has parts that are supported by the Move Operation it is also a consideration , with higher risk since the items are Moved not Copied!!!
Below is the Screenshot of the Subscription Change Directory operation, which has solved similar issues for me!
Just select the Link and the Directory and wait for a few minutes!
I hope this helps!
Kindly mark the answer as Accepted and Upvote in case it helped!
Regards