fix: Port OAuth fixes from interactive clients to automated clients #1057

Job	Run time
typescript / Check Typescript library	42s
typescript / Check Typescript E2E tests	34s
python / Check Python E2E tests	10s
typescript / Check Typescript chatbot	34s
python / Check Python library	43s
cdk / Check Python-based MCP Doc Server	1m 52s
cdk / Check Typescript-based Cognito stack	17s
cdk / Check Python-based Book Search Server	2m 12s
python / Check Python chatbot	13s
cdk / Check Typescript-based Weather Alerts Server	1m 33s
cdk / Check Python-based Inspiration Server	30s
cdk / Check Typescript-based Cat Facts Server	1m 14s
cdk / Check Python-based Dad Jokes Server	2m 13s
cdk / Check Typescript-based Dog Facts Server	1m 31s
cdk / Check Python-based Time Server	1m 54s
cdk / Check Typescript-based Dictionary Server	1m 51s
	18m 3s

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