Peter, I just posted pix of my semi-booth that also doubles as a winter clothing/linen closet in the thread on studio pictures started by guitarhacker.

This arrangement, where I have a non-enclosed 'booth', but in a very dead part of the room due to the fluffy clothes, sleeping bags, etc. and the acoustic treatment on the back side of the GOBO that I face into the closet, makes for a really nice recording space for vocals, acoustic guitar and other single or double mic recordings.

There's another advantage to this room - there is no HVAC ductwork in/out of the room. It's much quieter than the actual studio space I built in the basement of our previous house where tons of noise would come into the room through the forced air ductwork.

If I were you, I would experiment with whether to actually keep your door open or closed for your closet. Closing the door can 'complete' the wall, which can cause a standing wave propensity, that would fall to the bottom end of audible range with the door open - at least in that dimension of the closet. Use a mic with a high pass filter and you are all set. I think that you will have better luck keeping that mode out of your mic with keeping the door open.

Use the room acoustic mode calculator: http://www.bobgolds.com/Mode/RoomModes.htm

I estimated your closet height was 7', and I put in 4' x 4' into the calculator.

The first mode is floor to ceiling, at 80 Hz.

Next modes are the side to side modes, equal at 141 Hz.

If you open the door, that will kill off one of the 141 Hz modes for the most part because the wavelength will get much longer, not knowing how big your room is that your closet is in, but guessing that it's probably 10-15' across, you would have a 14' half-wavelength, or 28' wavelength, which is 40 Hz - if the room is 10' wide in that dimension. Add another 5' and the mode drops all the way down to 30 Hz.