Especially in a (mathematical) culture where "getting the right answers" is prized and excellence on such examinations is valued in general, it is reasonable that students will value this above other expressions of mathematics. (I have a little teaching experience in that context, though most of my experience is in the "just get through it" American culture of math.) So one way to get students interested in proof is to express something where the mathematical techniques are irrelevant and do not help in solving an interesting problem.
Ideally, you would be able to have students in a course like number theory or graph theory - both of which can be quite applied - and then give them a problem which simply doesn't have a (purely) computational answer. Like "is there an integer point on this elliptic curve", or "can I embed this graph on a computer chip with no crossings of circuits", or something. Maybe those are too advanced, but I hope you take my meaning. The only way to know for sure is to prove.
The difficulty is that most course content in this situation is more in the precalculus-through-differential equations trajectory, which doesn't always lend itself well to this. You can prove things like the intermediate value theorem or existence of solutions to equations, but students will not care about these. Because they seem obvious, like the fact there are no integers between $0$ and $1$ - you can prove it from some axioms, but students will not see the point.
But (perhaps ironically) in numerical methods in these areas, proofs are perhaps much more useful. How accurate is Simpson's rule? How much computer power do you need to get within $0.0001$ for your integral? Proofs are key to this - especially the somewhat (to students) bizarre requirements about absolute values of various derivatives. Show an example that shouldn't work, but it does. Now we need a proof.
Another idea is to talk about integration techniques. Give them a really hard integration by partial fractions or nasty multiple parts/substitution, then another. When does it work, when not? You can prove it - or that it isn't, Liouville's Theorem.
I can't guarantee even a little success; it's hard to find resources at times (my examples are probably half-baked), and you have a lot of headwinds, as is evident from your post. But in my experience, showing people what mathematics cannot do is a first direction that will motivate them to love math for its own sake, not just for the satisfaction of a good successful computation. We need both.