Add automated compatibility tests for common WooCommerce payment gateways [Investigation]

Description

WooCommerce Payments 3.2.0 shipped with a bug that affected stores using the PayPal Standard gateway. A fatal error would occur when viewing the payments settings page (WooCommerce → Settings → Payments).

The bug was resolved in this PR https://github.com/Automattic/woocommerce-subscriptions-core/pull/27

We should investigate whether it's possible to add automated tests to check that WooCommerce Payments is compatible with other common WooCommerce Payments gateways, e.g. Stripe, Braintree, PayPal, etc.

"Compatible" could mean a number of things, but at a minimum, the following should be tested:

No fatal errors or warnings when WooCommerce Payments is enabled alongside other payment gateways.
Checkout works as expected when WooCommerce Payments is enabled alongside other payment gateways.

Acceptance criteria

Investigate the possible approaches to add automated compatibility tests for common WooCommerce payment gateways (e.g. Stripe, Braintree, PayPal, etc.)
Decide on the best approach.
Create a new GitHub ticket detailing this approach.
Close this GitHub ticket.

If automated tests are not viable, we should consider adding manual compatibility flows to the critical flows and testing instructions pages.

Additional context

Incident report where this was originally proposed and discussed - paJDYF-2ON-p2

Automattic / woocommerce-payments