Change Sentry init to avoid polluting global scope

[noeldevelops]

While "rubber ducking" about Sentry 3rd party errors showing up as a team, we (re?)discovered that Sentry recommends we initialize our client in a custom way. Let's do it!

See Node client example here: https://github.com/getsentry/sentry-javascript/blob/a27652f09538e91d3c6c23f8eeadb75743067428/packages/node/src/sdk/index.ts#L104

Might need some discussion and further investigation to make sure we continue to catch all Unhandled exceptions with Sentry 🤔