bump-*-pr: handle HOMEBREW_NO_GITHUB_API=1 being set #26470

Job	Run time
syntax	1m 9s
formula audit	6m 9s
vendored gems	44s
docker	4m 35s
cask audit	2m 24s
tap syntax	7m 57s
test default formula (Ubuntu 22.04)	4m 13s
test default formula (Ubuntu 20.04)	5m 25s
update-test (Ubuntu 22.04)	53s
test default formula (macOS 13 x86_64)	2m 59s
tests (online)	7m 58s
test default formula (macOS 14 arm64)	1m 37s
update-test (macOS 13)	1m 12s
tests (generic OS)	6m 41s
tests (Ubuntu 22.04)	6m 13s
tests (Ubuntu 20.04)	6m 37s
tests (macOS 13 x86_64)	9m 41s
tests (macOS 14 arm64)	8m 45s
	1h 25m 12s

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