The Exchange.Manage permissions basically means impersonation. On its own, it does not give you any additional permissions compared to what the user already has in Exchange Online. Thus, you need to also make sure that the user(s) in question have been granted the corresponding roles within Exchange Online's RBAC model. For example, to run the Get-ServicePrincipal cmdlet, the user will need to be a member of one of the following:
[09:59:18][O365]# Get-ManagementRole -Cmdlet Get-ServicePrincipal
Name RoleType
---- --------
Role Management RoleManagement
Rinse and repeat for all cmdlets you want to run.