Prioritise Quota Limit Increase for Amazon Simple Notification Service (SMS)

Hello.

The service quota and technical support teams are supposed to be separate teams, so I don't think you'll get faster response even if you upgrade to a paid plan.
In the case of an emergency, it may be possible to speed up the process by contacting AWS support, but in other cases it is stated that support will not be prioritized.
https://docs.aws.amazon.com/servicequotas/latest/userguide/request-quota-increase.html

For adjustable quotas, you can request a quota increase. You adjust applicable quotas at the account level, or the resource level. Smaller increases are automatically approved, and larger requests are submitted to AWS Support. Larger increase requests will take time to review, process, approve, and deploy. You can track your request case in the AWS Support console. Requests to increase service quotas don't receive priority support. If you have an urgent request, contact AWS Support.