It's not nearly as close as you are suggesting. Let's see:
- Language exchange starts: https://www.girlgeniusonline.com/com...?date=20061129
- Language exchange interupted by wasp: https://www.girlgeniusonline.com/com...?date=20061201
- Lucrezia starts taking control: https://www.girlgeniusonline.com/com...?date=20061204
- Lucrezia taking part II: https://www.girlgeniusonline.com/com...?date=20061204
- Lucrezia loses control due to locket, Klaus tries to kill her: https://www.girlgeniusonline.com/com...?date=20061208
- Lars steps in, Klaus gives "kill them all" order because he is dealing with the Other and it may be the only chance he gets before she takes over: https://www.girlgeniusonline.com/com...?date=20061211
- Fight scene: https://www.girlgeniusonline.com/com...?date=20061213
- Lars' death scene; Zeetha engages Klaus: https://www.girlgeniusonline.com/com...?date=20061215
- Tableau: https://www.girlgeniusonline.com/com...?date=20061216
- While fighting Zeetha, Klaus realises something (last panel): https://www.girlgeniusonline.com/com...?date=20061218
- Ni Tok (surrender): https://www.girlgeniusonline.com/com...?date=20061220
It's true that Klaus doesn't have much time to process things, which is why it happens in the middle of the fight.
However it would make no sense whatsoever for Klaus to suddenly give a surrender offer to a random Skifandran under the circumstances. Even later he is concerned that Zeetha is out to kill Gil, and we know he has zero patience or mercy for anyone who tries to do that. It's obviously not because she's a Skifandran, because he was prepared to fight her. The logical reason is that he suddenly suspects that is daughter has been sent after him or Gil (or both), and he doesn't want to kill her.
(There may be other details - we don't know much about Skifander yet, and we know even less about what Klaus knows)