I think it is more useful to think of the trap bar as a squat variant than a deadlift variant. I would not replace deads with trap bar work if you are looking for similar results.
Before you give up on deads, are you using double overhand grip for all of your warmups? If not, you probably should. Also, you should consider switching your mixed grip every rep or two (or each deadlift session, if you prefer) to make sure that both sides get their fair share of work. I tend to switch my grip every rep on work sets.The asymmetrical stress on my shoulders/traps from mixed grip is becoming quite apparent on squats.