build(core): Do not mangle private methods used by Replay

[Lms24]

In Replay, we access some private methods from the core SDK package from the Client class. Because we mangle private fields by default, our SDK CDN bundles wouldn't expose these methods and fields anymore in their clear name to external callers, such as our Replay CDN bundle, producing runtime errors because the methods were not found.

This PR fixes these errors by explicitly excluding the called private methods from mangling so that they can be accessed externally. This is admittedly not a clean solution but it unblocks minified CDN bundle users. In the long run we can revert this PR once we streamlined Replay event sending into the SDK (#6480) (as well as breadcrumbs via hooks?).

Note: This unfortunately increases bundle size, not just for the Replay CDN bundle, but also for the Browser SDK CDN bundle.

fixes #6466

[mydea]

Very nice! I made a PR refactoring _withClient, so I think we can leave this in here - that one was relatively easy to refactor: #6494

[github-actions]

size-limit report 📦

Path	Size
@sentry/browser - ES5 CDN Bundle (gzipped + minified)	19.7 KB (+0.08% 🔺)
@sentry/browser - ES5 CDN Bundle (minified)	61.04 KB (+0.14% 🔺)
@sentry/browser - ES6 CDN Bundle (gzipped + minified)	18.49 KB (+0.12% 🔺)
@sentry/browser - ES6 CDN Bundle (minified)	54.59 KB (+0.16% 🔺)
@sentry/browser - Webpack (gzipped + minified)	20.27 KB (0%)
@sentry/browser - Webpack (minified)	66.25 KB (0%)
@sentry/react - Webpack (gzipped + minified)	20.29 KB (0%)
@sentry/nextjs Client - Webpack (gzipped + minified)	47.29 KB (0%)
@sentry/browser + @sentry/tracing - ES5 CDN Bundle (gzipped + minified)	26.68 KB (+0.07% 🔺)
@sentry/browser + @sentry/tracing - ES6 CDN Bundle (gzipped + minified)	25.13 KB (+0.08% 🔺)
@sentry/replay ES6 CDN Bundle (gzipped + minified)	41.85 KB (+0.03% 🔺)
@sentry/replay - Webpack (gzipped + minified)	37.89 KB (0%)