That’s not really a very typical or likely to work well use case for LoRaWAN and TTN, which are designed for data consumers that use a subscription mechanism over the Internet to receive the data collected from the radio nodes. The form of LoRaWAN currently implemented at TTN is not really designed to push data back down (“downlink”) to a different node.
Transferring data from one Arduino to another might be a use case for direct point to point LoRa however.
The LG02 is not really a LoRaWAN gateway, but rather more a node or a point to point setup doing a bad job of impersonating one.
An additional issue is that the Arduino UNO is a bit constrained in the processor’s memory resources to implement LoRaWAN - it can be done, but it’s not a good choice as you’ll spend a lot of effort dealing with limitations.
The TTIG is not perfect either…
You really need to first clarify what you are trying to do so that it becomes clearer if LoRaWAN or point to point LoRa is potentially a fit, then select suitable hardware.
For LoRaWAN that would probably be a more capable “super Arduino” (eg, something with a Cortex M0 chip and lots of memory rather than an ATmega) with a node class radio, and then a more traditional gateway consisting of an SX1301 or SX1308 concentrator and an embedded Linux host. Note that only this, and really only this connected to the TTN server infrastructure, is on topic on this particular forum.
For point to point LoRa it would probably be two “super Arduino” node solutions each with its own directly controllable node class radio.