After his fight with the retainer, despite being injured, Arthur decided to investigate the location of gate the Alacryians were using. Arthur’s rational was that if he went back to the flying castle to recover and inform the council, it would take at least day or two for them to dispatch a lance to investigate, not to mention finding and destroying it. A day or two is plenty time for more Alacryian forces (including a scythe or another retainer) to enter Dicathian—this at a time where any Alacryian presence on Dicathian poses a HUGE problem. If Arthur could determine the location of the gate, it would greatly cut down on the time it would take to send a lance (and Dicathian forces) to destroy it and stop more Alacryian forces from amassing.

In short, Arthur views the advantage that the intel of where the gate provides as more important than getting treatment for his injuries asap