Hope I'm in the right place...

Summer is here, which is bad news for me due to excessive allergies. I
literally cannot sleep with a window open or even use a window AC, as
even with the vent closed, it'll still manage to draw in a little
outside air....the pollens give me nocturnal asthma and I have so much
trouble breathing I can't even fall asleep. I lived in a home with
central AC last summer, but this is no longer the case.

Basically, I need a portable AC unit to cool a 13x20 bedroom that will
vent hot air either under a doorjamb or - I'm wondering if this is
even possible - down a heating vent. I see a ton of models on eBay and
locally and have no idea what to look for. I don't mind emptying a
water reservoir, but I simply cannot have the windows open, period. If
I can solve this problem for around $500, all the better. Any advice
is MUCH appreciated. Thank you.