I have a "soft locker" DL in a spare 4.11 center section. It works well and has decent street manners, nicer than a true DL. Has to do with the way they tuned the springs in the locker, it makes engagement/disengagement a little softer. It will still clunk - hey, that's a DL. For street tires at the track, a cone or gear type diff would work, you should be driving around the water box anyway unless you are running Drag Radials. Any serious burnout on street tires just causes the hard tread compound to ball up, the track rubber to ball up and you launch on bitty rubber marbles. For decent power leaves on DRs or slicks, locker or spool. And with a locker, make sure you drive straight to the line from the burnout. If you get sideways, straighten out, back up and go in straight to be absolutely sure the locker is locked.
Note - you can hook both tires on the drag strip with an open diff, just do two burnouts. Seriously - heat one tire, back up a little, burn - Tire 1 will stick and spin Tire 2, stage, go. Works.