double doomwheel and not double abomination? You're lucky your brother likes to play nice lol.
Now, people on these forums will nearly always give the same advice for just about any problem unit "just throw more skinks at it." and most of the time this is sound advice. Sadly, I'm not sure if they are aware that the doomwheel has a 4+ armor save and three ranged attacks at better range than the skinks.
A few things to remind your brother of that makes the doomwheel less scary:
#1: In ANY phase that the doomwheel takes a wound it must roll to see if it goes out of control. So a skink shot that lands on the shooting phase, a direct damage spell in the magic phase, and potentially a wound due to dangerous terrain in the movement phase will all cause an out of control test. But, not a melee strike in the CC phase as it specifically says that being locked in CC prevent an out of control test.
#2: I see this one a lot, make sure your brother knows that the 2d6 attacks the doomwheel makes in CC are at S2. It only gets d3 attacks at the full S6. If you look on the profile it specifically shows that the 2d6 attacks are coming from the crew, not the wheel.
#3: Zzzzap! bolts always target the nearest unit, friend or foe. So if he has a unit of clanrats closer than your units then he must target the clanrats. He can stop himself by making a leadership test but the doomwheel only has a leadership of 7. That is a 50/50 chance the bolts are going to hit his own units.
As an experienced skaven player there are a few things that worry me when fielding the doomwheel, which I will now share with you:
Cavalry: Not for the large AS, but the large movement. The doomwheel is actually a lot like our ancient stegadon. We can stay back a little and pelt with shots but we really shine when we land a charge and get all those impact hits. That is basically an identical scenerio. If you land the charge against the doomwheel instead of vice versa you take a lot of it's power away while laughing off the S2 hits.
Monsters: Lets face it, monsters are freakin brutal. What do Lizardmen have that would constitute a monster good for taking down doomwheels? Why a plain Jane stegadon of course. Bolt thrower shots for while you're closing the gap (which can cause the doomwheel to go out of control) and if you land a charge you deny him impact hits as well as putting down some impact hits yourself. If you are willing to pony up a bit more points than a skink chief on an ancient stegadon with the stegadon war-spear will obliterate a doomwheel on a charge.. but also cost roughly twice as much and eat hero points.
Warmachines: They hit hard and make things go boom. Again we are back to the stegadon, our only real warmachine as a moving bolt thrower.
So, in summary, due to the doomwheel's nasty multiple wounds (d6) from it's lightning I would suggest a combination of skinks to try and get some hurt on them with potential redirecting combined with CoC or, if you go lore of life, a big fat saurus unit that you bump to T6-8. Throw things at it that are a threat without letting him capitalize with his multi-wound attacks. Remember, each wound only affects one model, then said wound gets multiplied. If it strikes a model with only one wound then it can only take one wound and multiplying doesn't matter. Sending units of tough one wound models and shooting that can get around the high toughness will save the day.