That is a very small sub. There is a saying in speaker design, which says: - "Does a speaker have to be large? No but it really, really helps."
Then there is Hoffman's Iron man's law, which basically says that the smaller the speaker the less sensitive it will be, especially in the bass.
So at your specified volume that commits you to a sealed design. There are very few drivers available that can produce a sub in that volume. Because of the volume of the construction materials, the driver volume, bracing and plate amp volume you are looking at an internal air volume of under 1 cu.ft.
Now in a sealed enclosure the output is going to be dropping above the last octave. So that requires a driver with a massive motor system that can take the power required to force output below the Fs of the design. That is the frequency at which output is 3db below reference. Below that point the driver has to be able to take a boost at 12db per octave. For every 3db of boost you have to double the amp power. In order to prevent damage to the driver, you need a high pass filter to cut the driver output at the point the driver exceeds it maximum excursion, known as x-max. Now in such a design the internal air pressures are enormous, and so that cabinet has to be extremely rigid with figure of 8 bracing. Extreme rigidity is called for. The driver will need rolling off at 26 to 27 Hz range. I will know more if I do a model, but I don't want to spend time on that, unless this design truly appeals to you.
Now there is a driver that could be used in a design to your specification.
It is a Dayton Audio 10" Ultimax sub driver. It is robust enough to handle the power.
The driver and a suitable plate amp will set you back $509.00
By the time you have built that cabinet and finished it you will be at the top end of you budget, and likely a little over.
So, if you think this is the way to go, then I certainly could model it, and come up with a design.
However, in terms of performance it will not come close to competing with a much larger ported design.