Well like i said this is tricky, because of the many variables at play... its hard to find a fine line which makes it worthwhile without being exploited. Thats why im glad its not my job to find carrot and stick aproach that works.
Increased renown % for underdog... already there, maybe more but it won't change too much for casuals as increased renown if your kills are low to zero from being so outnumbered is not really worth. So people swap sides to ride gravy train to make easy gains/character progression.
Then can't give combat bonus to underdog because organised would take full advantage of such things (also some map borders with pve land can step out and so its open to be abused a little), also especially as sometimes you might have more numbers in a zone on one side but they are afk on BO so buffing other side and realistically you are outnumbered then facing extra buff's... so thats pretty counter productive.
As for wanting BIS from get asap, yeah thats what people aim for but shouldn't expect it straight away without some work... its not that hard to farm in a team enviroment also.
Making a reward that veterans want, as well as fairweather casuals is also tricky... to do something which is not broken.
Also making some loot for fighting with AAO ... people could just stand around afk in zone with aao buff
so if its possible... make it kill/contribution linked?
Perhaps something cool cosmetic skin or dyes linked to number of rvr kills during % AAO would maybe entice some more people to try play underdog more and try close the gap in population shifts, but at the end of the day it depends on players priorities... some will want easy gains/bags/rr for boring gameplay and not care a damn about balance or any fancy skin.
The perception of x-realm is the everlasting problem of 2 faction system, when one side dominates too much, other side give up hope and concedes, where if there was 3 factions logically the 2 weaker would try double team the strongest so things balance out, instead of by domination/numbers/boredom route.
Maybe another thing, give incentive rewards shiny skins/mount/dyes to pug wb leaders who have played alot during AAO? to encourage more leaders
A possible cosmetic NPC if you have fought enough contribution with AAO during a zone capture ... maybe you earn 1 token per a zone? and need 30-100 tokens to buy something shiny from npc? Whatever seems suitable and non gamed system
I don't know the perfect solution from dev or player POV, just trying to think of a incentive that doesn't break gameplay/exploitable and enough of a carrot to make people think about sticking out being outnumbered and fighting hard to the last... instead of abandoning their realm in defeat and jumping on bandwagon... that would help go with your plans of limiting traffic flow in the wrong direction so the bigger force is less likely to keep getting bigger creating snowball effect.
in a Ideal world players wouldn't need incentive to go to underdog and would just do it for the challenge and action... but the world isn't ideal, thats why carrot and stick exists.
I would like to think that if casuals know they are making some character progression they might stick out being underdog/ or farmed and atleast try to fight back... nobody likes to lose but if they believe they are wasting their time with no character progression then I see why some of them join gravy train or give up. If they lose but still make character progression, if rewards have enough flavour to them ...
Then its like scenarios where majority of people stick out 500-0 for 1 medal for losing, instead of ragequit at portal because they are working towards something. But 1 conq medal in Orvr as a reward is not big enough pinata to entice people for their time, for gear they probably already have. Something like more unique tier of skins or mounts if possible in warhammer universe might just be appealing enough without breaking dynamics of game ?
But generally this small community is under magnify glass, makes things like this seem bigger than they are in a two faction system... unless you know for certain organised groups consistantly do it intentionally time and time again then that would be a issue but i doubt it's that bad, its more a bit of game rage look for excuse of failing...
x realm is the easy go to blame game to vent on nameless people. People naturally get tired/demotivated if fighting superior numbers or organised forces, they have work early or can get more fun investing time in something else so give up earlier or easier due to psychology of war and no hope/confidence in their current realm locking zone / or if the reward is worth the extra effort to push on.