If you're asking whether there is any chance we'll be able to do this in the forseeable future then answer is no.
If, however, you're asking whether general relativity allows constructions like this then the answer is yes.
Your question refers to a brane which would exist inside a bulk of our own design, and I don't know how literally you meant this but it would have to refer to a 2+1D universe embedded within ours. That's impossible because there's no way to restrict any physical system to two spatial dimensions. What GR allows is geometries referred to as a bag of gold spacetimes.
I'll use an analogy to explain this, but please remember this is just an analogy so don't take it too literally. Suppose we represent our spacetime as a rubber sheet - you'll no doubt have seen popular science programmes using this analogy, even though in some ways it's a poor one. Draw a circle on the rubber sheet then inflate the sheet within that circle so it bulges out like a ballon. The result would look like the region labeled FRW region in this diagram:
(image from this paper)
If we get the geometry correct the the balloon will start expanding like our universe does, but it wouldn't be expanding into anything. All we would see is something that looks like a black hole and the new expanding universe would be hidden on the far side of that black hole. Indeed Lee Smolin has suggested that all black holes have a universe like this hidden on their far side, though I must emphasise that most physicists regard this as fun but unlikely.
But, but, but, but ...
The bag of gold spacetimes are constructed by welding together two solutions to Einstein's equation. Typically a region with the FLRW metric is joined to a region with the Schwarzschild metric. Mathematically this is a perfectly reasonable thing to do and the result is also a solution to Einstein's equation and therefore a possible spacetime geometry. However what this process does not tell us is how we would actually construct such a spacetime starting from flat space. As far as I know no-one has shown it is physically possible (or indeed impossible) to actually construct a new universe in this way.