Given the need to automate a welcome email including a customized survey with customization requirements and a tight timeline, the best approach is:
Use Salesforce Flow to build the survey declaratively to meet the criteria and send it to the customer as an email (B). Flow is a powerful tool for automation in Salesforce, allowing the creation of complex workflows and user interactions without needing to write custom code. It can be used to build a survey and integrate it with email services for automated dispatch.
Working with a developer to create custom Apex and LWC (A) might meet the criteria but would not align with the limited time frame. Managed (C) and unmanaged packages (D) from AppExchange can provide pre-built solutions, but they may not offer the needed customization or could require more setup time than building directly in Flow.
Reference for automating processes with Salesforce Flow:
Salesforce Flow: https://help.salesforce.com/articleView?id=sf.flow_considerations_design.htm&type=5