Well, first off, it's good practice to do as little non-UI work on the foreground thread as possible, regardless of whether the requests take a long time. You don't want the UI thread blocked.
This sounds like a good use case for RabbitMQ. Primarily because you don't need each message processed by more than one consumer. If you wanted to process a single message more than once (say for different purposes), then Apache Kafka would be a much better fit as you can have multiple consumer groups consuming from the same topics independently.
Have your API publish messages containing the data necessary for the third-party request to a Rabbit queue and have consumers reading off there. If it fails, you can either retry immediately, or publish to a deadletter queue where you can reprocess them whenever you want (shovel them back into the regular queue).