Paizo monsters rarely ever do anything so sensible.
Paizo tends to include 'This monster fights to the death' and will try to include a reason such as 'This creature has been bound to the confines of this room and fights to the death'.
Basically, it's their way of bypassing smart creatures using their abilities to escape. Realistically, a dragon should almost never be killed. They are very intelligent creatures, and often times very old. An Ancient Red Dragon may be cocky, but he didn't live to be such an age by being stupid.
Granted, younger dragons would probably make such mistakes, but that's all a part of weeding out the weak. The only time older dragons should be killed, is if they absolutely cannot afford to retreat, for whatever reason.
This is usually done by 'being bound to this location' or something to that effect.
In case of the Rakshasas, they don't want their secret to spread, so they fight to the death to protect it. I might not be the best motivation, but it will have to serve.
On the other hand, each GM can decide for himself how the Rakshasas act.
They also have a great situation in Korvosa, which they most certainly want to maintain. If the "Arkonas" had to go back to Vudra, they would be very low on the totem pole. If they went to some other city, they would be starting over completely. Plenty of intelligent people have fought against the odds and died to protect what was theirs.
The best case scenario after an escape from the PCs is that they can somehow help Ileosa defame them. Since the PCs are assumed to flee the city directly from Palace Arkona, they don't have a lot of time to get the family secret out. If Balor and Vimanda can aid in assassinating the PCs' characters they might be able to survive the PCs attempting to out them upon the party's return to the city. If anything, the Arkona family is forced into siding with Ileosa until the PCs are out of the picture, which could possibly have ramifications on the rebellion in CoF. Ultimately, they hate the Queen just as much as the PCs, so this is not desirable. Add this all together and you can begin to see why the Arkonas might push all of their chips in now and fight to the death.
All of that is just one GM's opinion, of course. Hope it inspires!
The Arkonas might also broker a treaty with the PCs to keep the secret from getting out. Keep in mind, as evil as the Arkonas are, they've done a lot of good for the people of Old Korvosa.
Just because they are evil, doesn't mean they can't be genuinely concerned about the state of the city. They aren't the evil villains who are trying to blow up the world (and themselves). They are the evil villains who want to rule over their little kingdom, but they don't want any harm to come to it.
Point out things like the Arkonas have sent out their own men to keep the peace and protect the people of Old Korvosa and are distributing supplies to help the people weather the storm. They may be the ultimate head of the criminal organizations in Korvosa, but that just means that the criminals are under control and aren't performing their deeds on whomever they feel like. It's the only legal guild in the city, because it is the one that is most needed otherwise you have a bunch of free-running criminals, like Gaedren Lamm, that will do absolutely anything to make a copper piece.
In case of the Rakshasas, they don't want their secret to spread, so they fight to the death to protect it.If they die their secret is revealed, so unless they know it has already been revealed it is a motive not to fight to the death.
I admitted that it wasn't the best of motivations. In my campaign I won't use the Rakshasas at all, just because this is too big a secret. The PCs will want to kill the Arkonas for being monsters and the Arkonas will want to get rid of the PCs because they know their secret.
My approach is to have Glorio be a master manipulator, who wants the PCs to work with (or even better 'for') him. If I make him a rakshasa, that will only be in the way of this plan. So I will actually go for Tel's suggestion to have Arkona try to broker a peace with the PCs.
Still, when I get into the logic of the adventure as written, I feel that the Arkonas will probably fight to the death to protect their secret. Of course their secret will be out if they die, but at least they died trying to protect it.