I would say no.

The desert is roughly laid in between the equator and the 20ºS parallel and that's the intertropical convergence zone, the good-old-ITCZ. It is basically THE zone for copious daily rain, no seasons and constant humidity.
(check this: ITCZ - An Overview of the Intertropical Convergence Zone or ITCZ or this image:http://upload.wikimedia.org/wikipedi...nuary-july.png).

Deserts are located further south (or north) close to the tropics (and not the equator). That would be around 25-35 degrees, (south in the case of your continent). This is where dry air from the upper layers of the atmosphere sinks (subsides) and heats up in the process (drying even further) and it's called the high pressure belt.
(if you want some easy reading, check this: Brief notes on Sub-tropical high pressure belt)
on
I agree with you on the lack of rainshadow effect. Typical winds, in the nothern part of that continent are East-West, warm and full of moist. In fact the general shape of the range would drive them on a long curve southwards and upwards, spreading the rain through the slopes. Depending on the altitude, I think it would be a great place to grow coffee .

So, to sum up, here's what I suggest:
- I would keep only the part in lighter orange as desert (give or take),
- establish a tropical forest or monsoon region in the equatorial area south of the main range
- separate the two with a line of savannah

All this is of course, assuming Earth-like behaviour. All this is of course, my feel-free-to-ignore rambling.