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What the courts focus on are whether or not the non-payments are 'willful' and intentional. That is key. Since you have evidence of his postings that describe employment, you can therefore show a reasonable narrative that his failure to pay for the child is no longer accidental but intentional. The fact that his taxes are not in order is a different issue--the courts generally won't get involved, but you can report him to the IRS. However to the judge provide direct print-outs from Facebook or other social media where the ex has made the comments himself, and if the opposing counsel objects, state that as these comments are 'made against interest', they should be admitted as evidence. This would be the best way to go, although be careful that if you print out someone else's comments about your ex, the attorney may claim 'hearsay' and state that as that person is not in court themselves to explain the comments, they may not be admitted. The best response to that is to counter that as the comments were made in the open, they are public record and it is up to the judge to evaluate their validity or authenticity.
To be honest, how he answers is his own problem. If you requested discovery and your attorney properly framed that request, he cannot really deny it. So at worst he can try to delay, and if does so long enough it creates a contempt claim. No, it is not a felony if he refuses to pay, it is still a civil issue, but failing to pay creates a basis for potential jail time and contempt until he does begin to pay. He is also at risk of losing his driver's license.