Constraint 1: Cost of remaining materials
Because I already have most of the materials from last year when I started thinking about doing this project, I don’t have a ton left to purchase. The three main things that I need are: power connectors ($14), bluetooth amplifier ($35) and wood (the cost is going to vary with the size of this project). Unfortunately, wood costs can balloon pretty quickly with a larger project. As a result I may try and design a somewhat unusually shaped enclosure that would save me material. I’m thinking that the enclosure could be trapezoidal.
Bluetooth amplifier board, image credit 
The AC connectors that I’m thinking about are shown in the image below, these will give a really clean look for the project, because it won’t have a cord dangling off the back.
Inexpensive AC adapter option, image credit 
Constraint 2: The woodworking aesthetic
Working with this aesthetic will certainly constrain my construction process and my material choice. I am going to be limited to things like table saws and routers. While the laser cutter is wonderfully simple and fast, the burnt edges that it produces really conflicts with the aesthetic that I am trying to go for. Also because I am going to try and do a natural wood look, I am going to need to choose decent materials, which can drive up the cost as well as make things harder to find. I was originally thinking about doing a patterned walnut and cherry face of the box, but that would have been fairly expensive as compared to pine. Therefore, I am just going to stick with pine or one of the other less expensive woods and either leave it natural or use a stain if I want a darker look.
Constraint 3: Time availability at the end of the semester
Currently I am building a robot in 2 of my classes, and those projects are going to get even more demanding as the semester wears on. I am also in a business class for the CU MBA program which is quite demanding. As it gets later in the year, the strain on my schedule will get even worse. Therefore I want to try and not bite off more than I can chew. My goal is to get started as soon as possible so that I can front load things and to make this project more realistic so that I don’t completely screw up my schedule.
Constraint 4: Size of the enclosure & fitting the maze
One constraint is the size of the project. I am living in a fairly small apartment and will be moving to one of similar size or smaller in August. Therefore the issue of size comes into play. On top of that, the larger the enclosure is, the more wood I have to buy, so keeping it to a manageable size is really important. On the other hand, the enclosure needs to be large enough to hold the speakers that I am planning on using and to contain the maze for at least one of the speakers for a bass boost. One more problem is that the larger I make this thing, the more difficult it will be to store in the shop or to transport around.
Constraint 5: Availability of shop space
Another issue related to timing is the fact that all fabrication spaces get progressively busier at the end of the semester. This means that if I leave myself only the last couple of weeks to work on the project, I am going to have to compete for space. And given that I have a very busy schedule, I may have a hard time finding periods of emptiness in the shops around campus. I am planning on using the wood-shop in the idea forge, so hopefully that doesn’t get too crowded.
Credits/ References: Digikey 1, Digikey 2  Parts Express
As I was reading through your post I realized how similar all of our issues actually are even if we are all doing different projects. Time, cost, and aesthetic seem to be the most important of your constraints you listed above. Fulfilling your natural wood aesthetic shouldn’t be too difficult depending on how you plan on fabricating it. If you are using a laser cutter I would look up different types of wood joints; they have some where the joint is actually hidden which might mitigate some of your concern for burned edges to be showing. Cost and time will both be dependent on the size of enclosure you chose to go with; I would suggest possibly building the electrical component and then measure and design the enclosure for how big you need it. Or if that is unfeasible you could laser cut paper enclosure as tests so you can see if the dimensions will be big enough before you cut your wood.
I know Bose has some speakers that use compact mazes (I think it is called the soundwave). If you are looking for inspiration on how to craft one I would recommend looking into them.
I also like how you are planning to go for a natural look with the wood. Every speeker system I have seen is black, and I think that your Idea will bring a nice ambiance to your apartment.
You mentioned a maze this needs to be big enough to enclose, is that just a feature of speaker enclosures? I recommend looking up “X Grilling planks” on Amazon, where X is the type of wood you’re looking for. You can get a decent amount of a specific wood this way on the cheap, although it’ll limit your dimensions for you. For example, the ones I’ve ordered are 6″ x 13.5″, although it’s slightly less if you want to chop off the end with logos. Alternatively, you could use the unbranded side of the wood, but it tends to not have as smooth a surface. You can always bring them to the manufacturing center as well for a quick sanding.