Disclaimer: Note again that this is not the approach I'm advocating. I don't even think it's a good approach.
It feels to me that you're putting yourself in Hazō's shoes and then modeling probable outcomes from that. ("I care very much about protecting the Condors from genocide and (not as much or not at all) about Jiraiya being Hokage. Jiraiya's Hokageship is fragile and my opposition would endanger it, therefore giving me leverage over him. I will tell Jiraiya that he must protect the Condors from the Pangolin or else I will oppose his Hokageship. He cannot afford to coerce me in any public way, so he will capitulate and do what I want him to.")
Would your estimated outcomes be any different if you put yourself in Jiraiya's shoes instead? Imagine yourself as Jiraiya, with your best understanding of Jiraiya's values. Your adoptive son comes to you and blackmails you to help the Condors. How does your simulation of Jiraiya feel about this? How much does he value the Condors and their genocide? How does that level of care rank in his other values? Are any of his other values involved in this situation? What is his most likely response?
(Quick clarification: Hazō would not be demanding Jiraiya's personal help, but permission for Hazō and Keiko to act.)
I don't understand the difference you expect from this reframing. Nonetheless, —
I don't think Jiraiya particularly cares about the summon realm at all except as an exploitable resource. Obviously his main priority in almost all regards is Leaf, with smaller, more private motivations on top, like for his own success. Jiraiya also values his personal agency, as you can tell from his discussion on going missing-nin; he considered the idea poor not because of the idea in general, but because he Leaf would send the big stuff after him, and he considered that a problem because he wasn't confident he could handle it, not because it would have hurt Leaf to spend the resources. Finally, he seems to claim he has some kind of specific attachment to the family, but I don't give that much weight overall since it doesn't make a ton of sense to me and he could just be saying it to increase trust.
I think Jiraiya will have two main concerns here. First, and perhaps most trivial, the reduction of exploitable resources from the summoning realm. I expect Hazō would try to compensate by offering some kind of mitigation if possible. Second, and the one that bites, a loss of trust in Hazō generally; Jiraiya presumably categorizes people a lot by whether he considers them ally or adversary to Leaf, and if Hazō doesn't follow that incentive structure Jiraiya will find it hard to consider him an ally. If Hazō is sane (not, admittedly, a given), he will make sure his incentive structure is clear, reassert his fealty to Leaf in other respects, and assure Jiraiya that he is in other ways predictable—as in, Hazō has no intention of repeating this performance, and considers himself in political debt.
There are two reasons I think Jiraiya would go along with it. The first is that I don't see how Jiraiya can afford to say no; the value proposition would be atrocious. Fundamentally—if he cares about his post as Hokage in a fundamental way and, from what I've given above plus his claims about the fate of Leaf if he fails to get it, I expect he does—he doesn't have all that much choice. This doesn't prevent him making Hazō's life difficult in return, or retaliating more fully after-the-fact, but the former is assumed acceptable in this respect, and any after-the-fact actions would be moderated by the fact that Hazō would be making good on his guarantee to continue working in Leaf's interest.
The second reason is more outside-view level stuff. "I won't support you unless you do X" basically summarises Leaf politics in a nutshell. This is how the game is meant to be played, and if Jiraiya was the kind of person to play hard lines against such tactics, he wouldn't be successful. Jiraiya is also capable of working with people who want different things to him, as his successful spy network makes clear, and refusing to ever compromise on things is a kind of stupid that, while it might work a few times when you're lucky, doesn't work consistently and certainly doesn't scale. So, again, I expect Jiraiya to say, "screw you kid, I thought we were a team" but then go on and let Hazō do his thing anyway.
