You don't have to modify the texture.cfg, just copy it from the FSDT_Staircase_USER\texture.FSDT folder.
I did exactly the same steps as you did, and it works, this is my texture.cfg in the texture.WWA folder, which contains the GSX_Logo_4x1.dds
[fltsim]
fallback.1=..\..\FSDT_Staircase\texture.01
fallback.2=..\..\FSDT_Baggage_Loader\texture.01
fallback.3=..\..\FSDT_Marshaller\texture.FSDT
And this is the sim.cfg for the FSDT_Staircase_USER
[fltsim.0]
title=FSDT_Staircase_01_USER_FSDT
model=01
texture=FSDT
couatl.icaoprefixes = LI
[fltsim.1]
title=FSDT_Staircase_01_USER_WWA
model=01
texture=WWA
couatl.icaoprefixes = KSEA
[General]
category=GroundVehicle
[contact_points]
wheel_radius=1.448
static_pitch=0.0 //degrees, pitch when at rest on the ground (+=Up, -=Dn)
static_cg_height=-100.0 //feet, altitude of CG when at rest on the ground
[DesignSpecs]
max_speed_mph = 5
acceleration_constants = 0.3, 0.4 //Time constant (effects responsiveness), and max G acceleration
deceleration_constants = 0.2, 0.4 //Time constant (effects responsiveness), and max G acceleration
[Couatl]
type = Staircase
cruise_speed = 2 // meters/sec
acceleration = 0.2
front_wheels = 2.2
rear_wheels = 1.65
condition = %(exitAlt)f >= 2.606
animated = 1
anim_lift = 0 100 10
interpIntervals = 0,0.5,1
extension = 3.675,5.809,7.942
height = 2.606,3.827,5.049
extensionAdjust = 0
heightAdjust = 0
// safedistance = 0.7 // distance from aircraft in meters before raising stairs