Bug bounties work, why wouldn't "feature bounties" also work?
You say you want those features, preferably bug-free, for this deadline. And there's $5K for the team if the objectives are met.
Then, your metrics problem boils down to how to impartially measure customer satisfaction, or how well the objectives are met (in some contexts, bugs are unavoidable etc.).
Metrics can still be important to help the team identify their problems (or rather, confirm that their intuitions about the problem). It's an optimization problem: measure first, then do something about the actual bottlenecks.
That said, some programmers are such nerds that more money is not the highest motivation. One can use some creativity here.
In this case, you'll get your features, but they very likely won't be bug-free. They'll probably be quite slow and fragile. They might not scale. They might not be well thought-out. They might break backwards compatibility, or break other features that your customers are already using.
In other words, why wouldn't a developer borrow limitless technical debt in order to claim the bounty as fast as possible and move on to their next bounty?
On top of that you have the problem that now your developers will follow the set objectives to the letter even if it transpires that something else was required.
> Measuring developer outputs can be detrimental
And then:
> Design and coding: The number of design papers and specs, work items, pull requests, commits, and code reviews, as well as their volume or count.
All they do is add more and more metrics. But this has the exact same problem as with the infamous KLoCs measure: how do you interpret it? How do you know it is not gamed, to begin with? Actually, now you have two problems: collecting and analyzing this mass of metrics can have a significant cost.