I moved my first batch outside last year on April 1st. They were 5.5 weeks old and I was over them being inside. So I evicted them to a yet unfinished coop. First night, had a heat lamp in there for them but every time I crawled out of my nice warm bed (and it was often, believe me) to check on them they were cuddled in a warm pile near the pop door - nowhere near the heat. It was 20 degrees and still dropping when I finally collapsed from exhaustion and went to sleep. Next morning they were just fine. Same thing the second night, except I only got up once. Third night the heat lamp came out and it snowed. And snowed. It snowed off and on until June 6th. In the meantime they just kept growing and thriving. So I started re-thinking the conventional wisdom of this temp for a week, then that temp for another week.....and keeping them in the house until they laid their first eggs in the brooder.
Now I brood outside from the start, and it doesn't matter what the temps are out there. As long as they have their heating pad cave they are just fine. (Like @azygous I was happy to see that you used that system.) By 4 weeks old and with temps up to the 30s, they were already totally integrated with the rest of the flock, no longer using the cave or the pad. We need to take care of them, absolutely, but they are not the delicate little divas we sometimes think they are. They are tougher than they look, and they actually do better with regular day/night cycles and regulating their own temperature needs. Remember that by this age they all wouldn't fit under Mama Hen anymore either...she'd be teaching them to roost. So take the cave system and rig it outdoors. It'll work just as well there. Here are some ideas to help you figure out how to rig that. You can probably turn the heat way down on the pad, even outside, and they may use it more for familiarity than warmth at first, but they'll be really happy to have space to explore! You've got this!!
https://www.backyardchickens.com/a/yes-you-certainly-can-brood-chicks-outdoors
Now I brood outside from the start, and it doesn't matter what the temps are out there. As long as they have their heating pad cave they are just fine. (Like @azygous I was happy to see that you used that system.) By 4 weeks old and with temps up to the 30s, they were already totally integrated with the rest of the flock, no longer using the cave or the pad. We need to take care of them, absolutely, but they are not the delicate little divas we sometimes think they are. They are tougher than they look, and they actually do better with regular day/night cycles and regulating their own temperature needs. Remember that by this age they all wouldn't fit under Mama Hen anymore either...she'd be teaching them to roost. So take the cave system and rig it outdoors. It'll work just as well there. Here are some ideas to help you figure out how to rig that. You can probably turn the heat way down on the pad, even outside, and they may use it more for familiarity than warmth at first, but they'll be really happy to have space to explore! You've got this!!

https://www.backyardchickens.com/a/yes-you-certainly-can-brood-chicks-outdoors