Well, specifically what he said was
which, to me, reads less "I'm not going to throw my warriors away on a fool's errand" and more "He calls himself a king but can't even beg properly?"
He likes him enough now, but, ironically, dwarves don't seem very sympathetic to people who've fallen on hard times.
That's... a way to read it I suppose
But I'm pretty sure he wasn't saying "I want him to beg" and more saying "He's sending some Umgi messenger to demand troops from me isn't he?"
Kazador was basically bitterly writing off Belegar, who mind you he didn't know at all at the time, as some jumped up claimant to the throne who probably wanted him to send his people to some meat grinder for a lost cause
King Kazador snorted. "The latest claimant? Is he sending manlings to perform his tour this time, instead of coming in person to ask to be gifted a Throng?"
Emphasis on these two specific portions
First, it sounds like Kazador has seen the whole song and dance before about needing support to retake 8 peaks, quite a few times actually, maybe there's been overly ambitious and overconfident Dwarves badgering him about it in the past, going on about some glorious reclamation
Point is he's heard the sales pitch before and it didn't work out so well then, so he's really not in the mood to hear it now
Second and related to the first, he's being dismissive because he has already decided in his mind in that one instant that whoever is claiming the throne of 8 peaks is probably some arrogant blowhard whose full of hot air
"ask to be gifted a Throng" he's got the immediate expectation that this so called king is going to demand
Kazador's people serve his interests out of some entitled belief that they're just going to help him with no regard for their own well being
Of course Belegar's letter proceeds to blow these expectations out of the water and immediately change Kazador's mind in the span of heart beats, but honestly I can see why he wasn't immediately excited about anything until he realized Belegar's expedition actually had some real teeth to it
And you have to keep in mind he was completely cut off from the outside world and any information of what was going on for quite some time