AWS IoT integration with The Things Network (TTN)

Based on the available information, there isn't a direct integration between AWS IoT and The Things Network (TTN) specifically for performing FUOTA while keeping your devices on TTN.

AWS IoT Core for LoRaWAN does provide comprehensive FUOTA capabilities as a fully managed LoRaWAN network server (LNS) that includes Firmware Updates Over-The-Air functionality. This service supports multicast FUOTA based on LoRa Alliance specifications, allowing you to update firmware on multiple LoRaWAN devices simultaneously using multicast groups. This implementation follows the LoRaWAN Remote Multicast Setup Specification, LoRaWAN Fragmented Data Block Transportation Specification, and LoRaWAN Application Layer Clock Synchronization Specification.

While you mentioned that TTN AWS IoT integration allows you to receive uplinks and schedule downlinks via the AWS MQTT client, there's no indication that this integration extends to supporting the complete FUOTA process while keeping your network server on TTN. The FUOTA functionality in AWS appears to be specifically tied to using AWS IoT Core for LoRaWAN as your network server.

To leverage AWS's FUOTA capabilities, you would likely need to migrate your devices and gateways to AWS IoT Core for LoRaWAN, which you've indicated you want to avoid for now. The documentation doesn't suggest a hybrid approach where you could keep your devices on TTN while using AWS specifically for FUOTA operations.
Sources
Perform firmware update over-the-air (FUOTA) for LoRaWAN devices and multicast groups - AWS IoT Wireless
AWS IoT Core for LoRaWAN - AWS IoT Wireless