Math’ed this one out.
*Assume +10 +res Cain, +10 +atk bike, 2 eirs, light season.
*Assume they get doubled by everyone and do not one shot. This advantages Bike artificially on both counts (Cain is very capable of one-shotting many glassy ranged threats), but some assumptions are better left unsimplified. Am definitely not unnerdy enough not to add a nth axis, but am lazy boi. This is supposed to make it better to be lazier in AR, after all.
*assume Brave Lucina and Brave Ike are not a duo unit.
*x axis is opponents atk stat, y axis is dmg taken over one combat (two hits), purple is cain, blue is bike.
case 1: enemy hits defense
Bike takes much less damage than Cain for enemies who hit defense that have atk >~60, taking only half as much for enemies in the 65-75 atk range.
case 2: enemy hits res
We see that around 65 atk, Bike starts to take less damage than Cain, although it is pretty close in the relevant 65-75 atk range.
The reality changes a ton when you add support in form of buffs, however. These benefit Cain much more than they do Ike at the same numbers.
Assuming +13 buffs from allies (combined in combat and visible), enemy hitting def:
Ike’s advantage still exists in the important 65-75 range, but is much smaller.
For hitting res:
Cain actually has the edge against units hitting res for much of the relevant range, but very slightly.
The point where Cain truly has the edge is when you can stack 20+ def/res from support on each, 13-18 being close. a minority of threats won’t double either Cain or Ike, although the vast majority will. More importantly, spd scales much better on Cain. As Cain gets faster and prevents more natural doubles, Cain is benefited much more than Ike. Stacking 50+ spd on Cain and he is taking far less damage than Ike in every situation except firesweep.