I've never listened to the Grado RA-1 but do use the HeadRoom Total AirHead for portable listening. The circuit they include to compensate for the fact that a pair of speakers would produce interference at each ear is intended to improve the spatial characteristics of headphones. The other effect is that it takes the edge off the top end since that is where the interference has its greatest effect. -- Though I used a tube headphone amp for a while (non-portable), I went back to the HeadRoom equipment (Little More Power Premium) because of much lower levels of listener fatique. -- I've also used the Senneheiser HDC451 active noise cancelling headphones in noisy settings. They don't have a lot of bass but avoid the difficulty of having to crank the volume to get a decent signal to noise ratio. -- Another, more costly alternative, is the Etymotic Research ER4P (louder than the ER4S with a roll off in the highs). Though they give you about 25dB noise reduction, they block outside sounds to a point that may not be appropriate for driving. -- Finally, having experienced the same effect with headphones when used with Silver Bullet interconnects, I have used inexpensive copper cables from bettercables.com ($60 for 1m). I agree that copper is your best choice.
-- Good listening.
-- Good listening.