fix(ui): dont mutate searchparams

[Skn0tt]

Closes #38265. I had trouble reproducing it in a test, but writing onto the Context value is never a good idea, so let's always use a copy.

[github-actions]

Test results for "MCP"

2432 passed, 116 skipped

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Merge workflow run.

[github-actions]

Test results for "tests 1"

1 failed
❌ [playwright-test] › runner.spec.ts:118 › should ignore subprocess creation error because of SIGINT @macos-latest-node18-2

4 flaky

⚠️ [firefox-library] › library/inspector/cli-codegen-1.spec.ts:1079 › cli codegen › should not throw csp directive violation errors `@firefox-ubuntu-22.04-node18`
⚠️ [playwright-test] › ui-mode-trace.spec.ts:397 › should work behind reverse proxy `@macos-latest-node18-2`
⚠️ [playwright-test] › ui-mode-trace.spec.ts:659 › should indicate current test status `@macos-latest-node18-2`
⚠️ [playwright-test] › ui-mode-trace.spec.ts:659 › should indicate current test status `@windows-latest-node18-2`

40316 passed, 787 skipped

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Merge workflow run.