With the bike I think this would be a solid Dredd win.
There's no reason for Dredd to be on the bike, he'd probably be better off using it as a diversion, or for covering fire, or order it to wheel around cap to facilitate some kind of flanking maneuver.
Cap ordinarily being able to dodge gunfire doesn't really mean anything concrete here either, Dredd is not a normal crook, he has pretty much superhuman skills with a gun (able to draw aim and fire his gun in a ludicrous 1/600th of a second according to one story), unless Cap is like Neo I think he would have difficulty here, Dredd could cover the whole area in well-aimed bullets in a second.
The Lawgiver is also a versatile weapon, If Cap uses his shield Dredd could switch to Ricochet and alternate between frontal and bounce shots.
If Dredd swaps to Hi-Ex and Cap blocks it shouldn't he be thrown backwards anyway? that would pretty much lead to him being dead too.
I'm not really even sure how it would end in straight hand to hand, Dredd is pretty damn resilient, he's taken multiple super-human level punches before, and was skilled enough to fight batman to a draw (though honestly I think Captain would take the majority here, his stamina is too good and he's most likely noticeably stronger than Dredd).
The biggest danger is the shield throw I think, that could well take out Dredd decisively, and quickly, though a miss, or serious but non-fatal injury on Dredd would probably result in Cap being dead (and heck, the bike providing cover could probably cut the feasibility of using this attack anyway).