test: Update profiling e2e test to use typescript

[AbhiPrasad]

Should catch issues like #12564

[github-actions]

size-limit report 📦

Path	Size
@sentry/browser	22.22 KB (0%)
@sentry/browser (incl. Tracing)	33.31 KB (0%)
@sentry/browser (incl. Tracing, Replay)	69.09 KB (0%)
@sentry/browser (incl. Tracing, Replay) - with treeshaking flags	62.41 KB (0%)
@sentry/browser (incl. Tracing, Replay with Canvas)	73.14 KB (0%)
@sentry/browser (incl. Tracing, Replay, Feedback)	85.31 KB (0%)
@sentry/browser (incl. Tracing, Replay, Feedback, metrics)	87.17 KB (0%)
@sentry/browser (incl. metrics)	26.5 KB (0%)
@sentry/browser (incl. Feedback)	38.42 KB (0%)
@sentry/browser (incl. sendFeedback)	26.85 KB (0%)
@sentry/browser (incl. FeedbackAsync)	31.42 KB (0%)
@sentry/react	24.96 KB (0%)
@sentry/react (incl. Tracing)	36.35 KB (0%)
@sentry/vue	26.32 KB (0%)
@sentry/vue (incl. Tracing)	35.16 KB (0%)
@sentry/svelte	22.35 KB (0%)
CDN Bundle	23.41 KB (0%)
CDN Bundle (incl. Tracing)	35.04 KB (0%)
CDN Bundle (incl. Tracing, Replay)	69.17 KB (0%)
CDN Bundle (incl. Tracing, Replay, Feedback)	74.34 KB (0%)
CDN Bundle - uncompressed	68.76 KB (0%)
CDN Bundle (incl. Tracing) - uncompressed	103.61 KB (0%)
CDN Bundle (incl. Tracing, Replay) - uncompressed	214.08 KB (0%)
CDN Bundle (incl. Tracing, Replay, Feedback) - uncompressed	226.67 KB (0%)
@sentry/nextjs (client)	36.23 KB (0%)
@sentry/sveltekit (client)	33.95 KB (0%)
@sentry/node	112.98 KB (-0.01% 🔽)
@sentry/node - without tracing	90.39 KB (0%)
@sentry/aws-serverless	99.48 KB (-0.01% 🔽)

[JonasBa]

Thank you @AbhiPrasad!