How to run standard flows of Order Management for Unmanaged Order Summary records?

We have a field on Order Summary object as ‘OrderLifeCycleType’ which is populated once when we create Order Summary record. Depending upon that,  Order Summaries are classified as  Managed and Ummanaged Order Summaries. 

Managed Order Summary has some limitations that we cannot edit the standard fields of Order Product Summary (Order Summary Line items) such as Quantity Canceled, Quantity Fulfilled, Quantity Returned etc, depending upon which Status is going to be changed. We want Status field to be changed to ‘Fulfilled’ so that the standard flows for Cancel items , Return Items etc should work. 

As these fields are not editable in Managed Order Summaries we decided to go with Unmanaged Order Summaries. So, now in Unmanaged Order Summary we are able to update the standard fields so as the Status. But now we are facing the problem that standard flows are not working for ‘Unmanaged ’ Order Summary , it only works for ‘Managed’ Order Summary records.

So, the conclusion is that

1. Either if we go ahead with Managed Order Summary records then we are not able to update the ‘Status’ to ‘Fulfilled’ . If it is not fulfilled then the flows will not work.
2. Or If we go with Unmanged Order Summary then also the flows will not work.

So, In both the case we are not able to run the Standard flows.

Will anyone please suggest the solution for the above issue?