Does the questioner's act of delaying the prayer—in order to perform ablution and relieve himself in a nearby mosque instead of inconveniencing the host—constitute an intentional delay of the prayer beyond its prescribed time and a departure from the religion?

"Prayers should not be delayed until their time is almost over. Your prayer is valid as long as it is performed within its prescribed time. However, delaying the 'Asr prayer without a valid excuse until the time of 'Isfiraar (when the sun turns yellow, indicating the approach of sunset) is considered a sin by many scholars, and one must seek forgiveness for it. Delaying prayer beyond its prescribed time, for example, until after Maghrib, is not considered disbelief by the majority of scholars, even though it is a grave sin."