Its called an octave. So its in the proper key, but an octave too low.
You wanted the Bb to sound higher than the Eb.
And indeed, this depends on the instrumentation.
These have a Bb that sound higher than Eb:
America
Mando Up
1/2 Step 1 and 2
Bowleg 2
And then there's some with lower bass but higher high notes, like Back and Forth, Gallop 1 and 2, etc.
It makes sense tho, a Bb is usually played in 1st position and Eb in 6th, which ofcourse sounds higher.
The exceptions are those with alternative voicings (ie. Back and Forth: Bb 688766 - Ebxx1343) or with the guitar tuned a 1/2 step down (Bowleg, 1/2 Step, etc). So i suggest you also try raising your song half a step (A - D - E - B) coz then most B's will sound higher than the E
