Melanie, I have brugs in some sun, and partial shade and they definitely do best in the partial shade for me. But in the full, blazing sun I think a young one would have trouble getting established unless you watered it every single day. They are heavy feeders and love lots of water and rich soil. Once it gets established you could prune it to keep the top growth thin enough that it could support itself out there in the full sun. But I'd sure wait until it has a big root system to transplant it.
Then there is also the issue of nematodes. Every brug I have put in the ground has sort of petered out after a few years and when I pulled them out, the roots were all knotted. That beautiful one you have might be planted near enough to the oak trees that the nematodes don't like the soil there. But unless I sink a big pot for them, mine have a struggle getting going near my oak trees, too because of all the root competition.
So, my advice would be to grow the little new one in a big pot at least until the rainy season begins, and it will get some cloud cover in the afternoons. Enrich the soil as much as you possibly can, to help with moisture retention and nematode prevention, then plant it out in June or July or August and water as often as you can in addition to the sprinkler watering. Show us a picture of it before you plant it out, and we can advise you on pruning it, too. (if it has branched by then)
As far as soil enrichment, if you can possibly handle a bag of alfalfa pellets (horse food, available at feed store or Tractor Supply in 50lb. bags for about $15) they are by far the best thing I've found for my garden in recent years. I sprinkle handfuls of them around everything, and even put them in my Earth Boxes and potted plants. The brugs just jump up and turn dark green whenever I add alfalfa - I've taken to sort of 'stewing' it in my old watering can, and then pouring the porridge of alfalfa over the top of the soil in the pots so they get a shot of dilute goodies right away, as well as the long term breakdown of the pelleted alfalfa.
