1. In the Event dialog, I'll add the Club Name next to the Member's name as you've requested.
2. You are correct to go first to the Inventory list and open the relevant item representing the Yellow Participation Patch (item 71458 I assume). Under the Club Usage tab, be sure the first checkbox is checked to indicate this is an item used in your club. Also check the 2nd box to indicate the item is being used as an Award.
Note: Set the main Inventory List to show "All items" otherwise you will not be able to locate items marked as 'not in use' since they will not be displayed in the list when showing only "Items we use".
After making sure the Inventory item record is good, save and close it. Go to Admin > Awards > Event Awards. Select the Event Name to show all currently associated awards. If the specific Award is not listed, add it by clicking in the top blank row that says, "Click here to add a new row".
[Note: in the Windows app, the Awards tab is under Admin > System Settings.]