[examples] Update APIGateway example with the new APIGatewayResponse initializer #396

Job	Run time
Semantic Version label check	4s
musl	1m 35s
Unit tests / Construct unit test matrix	3s
Integration Tests / Test archive plugin	2m 9s
Integration Tests / Test Examples/APIGateway on 6.0.1-amazonlinux2	30s
Integration Tests / Test Examples/BackgroundTasks on 6.0.1-amazonlinux2	30s
Integration Tests / Test Examples/HelloJSON on 6.0.1-amazonlinux2	29s
Integration Tests / Test Examples/HelloWorld on 6.0.1-amazonlinux2	29s
Integration Tests / Test Examples/S3_AWSSDK on 6.0.1-amazonlinux2	30s
Integration Tests / Test Examples/S3_Soto on 6.0.1-amazonlinux2	30s
Integration Tests / Test Examples/Streaming on 6.0.1-amazonlinux2	34s
Soundness / API breakage check	1m 41s
Soundness / Documentation check	1m 32s
Soundness / Broken symlinks check	2s
Soundness / License headers check	5s
Soundness / Unacceptable language check	2s
Soundness / Format check	35s
Soundness / Python lint check	5s
Soundness / Shell check	30s
Soundness / YAML lint check	5s
Swift 6 Language Mode / Swift 6 language mode	1m 14s
Unit tests / Unit tests / Linux (6.0)	1m 33s
Unit tests / Unit tests / Linux (nightly-6.0)	1m 33s
Unit tests / Unit tests / Linux (nightly-main)	1m 32s
	17m 52s

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