Yes Bm is the relative minor of D. In this case I believe I am playing in the B Dorian mode which uses the same scale as A major with a focus on Dm. Maybe I should change the title to D Dorian. The chords in B Dorian are Bm C#m D E F#m G#* A
In one sense you could call it the key of F#m. You‘re starting on the iv chord, and never playing the i chord, but that’s okay. There’s no rule that says you have to start on the root chord of a key signature or even play the root chord of a key signature in any given composition.
If you were writing this in standard notation you would have to give it a key signature. If you used Bm, every instance of the E chord would have an accidental (the G#). If you used the key signature of F#m, there would be no accidentals.
Now, I realize there is a bit of disagreement among some opinions on the internet about how to treat this, but I believe an understanding of what a mode really is helps clear it up.
A mode is not something unique and separate from a scale. It is simply a different starting point on the scale; it is a
mode of that scale. The key signature provides information about that scale, regardless of where you start playing it.
Keep in mind, every key signature has seven modes. We already express two of those modes with a common key signature. The major key is the Ionian mode and the minor key is the Aeolian mode, but it’s the same key signature.
I realize the dilemma you’re considering. The piece does not have its tonal center in either F#m or A major, so there is the temptation to call it something else. But, regardless, B Dorian is still just the second mode of the same scale that is expressed by the A or F#m key signature.
If you were going to change the title you should call it what it is: B Dorian. If you were going to write this in musical notation, you should use the same key signature as used for F#m or A major and make a note in the music that it is B Dorian.
Don‘t call it D Dorian because that isn‘t what it is. D Dorian includes the exact same notes as the C major scale (C Ionian).