How about no reward at all?
Temporarily drop the rep limit to 2,000.
Provide notes on what is expected of a Close Review.
Allow people who have a mind to to just plough into them (in a considered way, after reading the notes provided).
Nothing towards badges. Nothing towards anything. Close Review returns to 3,000 once some manageable number are outstanding.
No incentive for robo-anything. Simple mistakes in review are reduced in effect through requirement for multiple votes anyway. The standard of the notes provided (pops up when hovering somewhere, or a click, or whatever) reduces the number of simple mistakes.
If going ahead with any strategy, some investigation into what the major parts of the problem are, and which type of questions (if such can be decided) should be tackled first would be a big help, and motivational to see the length of the queue start dropping at the beginning of the task.
And not to forget, analysis of how it came about in the first place, and remedial action on that. I've seen a comment that the backlog existed, just was not exhibited, before the Review Queue system existed. Fair enough. But that does not explain 25,000+ increase in the last eight months or so. Or does it?
Without something to stop it happening again, it'll happen again. Then expecting The Community to do anything about it will be really demotivating.