Multi-Partner Management = True > Transactions Saved search (Opportunity, Estimate, Sales Order) > List all Partners under each Transaction > "Primary Partner" is not available to include as column

In the Transactions saved search Results tab, Primary Column is not included in the dropdown list. There is no way to drill down the Partner Fields to include it.

Partner gives the Primary Partner name. Partner Team Member lists all partners, if there are multiple partners in the transaction record.
If a partner is the Primary Partner, it is marked in the check box, or Yes in the System Notes.
If the partner is equal to the Partner Team Member, that means that it is the Primary Partner.
Below are the steps to list the partners and see if it is a Primary Partner or not, in each transaction record:
 
1. Reports > Saved searches > All saved searches > New > Transactions

2. Click the Criteria tab. Choose the following:

Type = is Sales order
Main Line = is Yes/ True

3. Click the Results tab. Choose the following fields:

Internal ID
Partner
Partner Contribution
Partner Team Member
Formula (Text) > Formula = CASE WHEN {partner} = {partnerteammember} THEN 'Yes' ELSE 'No' END  ......(*note: Custom Label = Primary Partner)

4. Click Save and Run.

5. Sort the list per Internal ID.

6. Sample results shown below:

Internal ID__Partner__________________________________Contribution__Partner Team Member_______________________Primary Partner
1494______102111Texas, Oklahoma, Arkansas, Louisianna __50.0%______43111 Mountain Shades - House Account________No
1494______102111 Texas, Oklahoma, Arkansas, Louisianna __50.0%_____102111 Texas, Oklahoma, Arkansas, Louisianna __Yes
(*in this list, the Primary Partner is "Texas, Oklahoma, Arkansas, Louisianna" which is correctly marked "Yes" in the list.

7. Click Export-CSV > Open
 
You may now make use of this in importing the data via CSV, to update the partner information of transaction records.