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When you’re adding a row that contains a linked field, you need to give the entire linked row as the value for that field, not just a portion of it. In your case you’re giving it just the client name, not the entire client row.
To fix it, you’d want to use the alternate form for populating the dropdown items where you give each item both a visible value and the real value. See this section of the docs for how to do that: Anvil Docs | Dropdowns and Data Tables
After you’ve done that, when you get the selected value from the dropdown you’ll be getting the entire row, not just the name. From there everything else should work.