Spraying latex is a non-trivial pursuit, and you want to do it indoors,
yet! Excellent chance of getting paint all over hell, and doing sub-opt
job. IMHO, you gotta be nuts.
What I*'d do:
1) dismount doors & drive a couple of long finishing nails maybe 1/4"
into top & bottom of each door ~1" from ends of edge.
2) paint each door, 1 side, with roller with door horizontal on horses.
Let each dry a few minutes before removing from horses and setting
close to vertical alongside wall, standing on 2 nails and having 2
nails on top in contact with wall.
3) when first side dry, place door horizontal on horses with first side
down, with nails only contacting tops of horses. Paint the rest. Allow
time to dry partially, then prop up to finish drying as before.
4) pull nails and hang doors.