How do I remove tetrabutylammonium perchlorate from an organic synthesis?
It's tempting to use phase separation with an aqueous phase, but you're clearly having issues.  One option is to evaporate the volatile solvents off, add toluene, cyclohexane or heptane and try water or another additive to boost the ionic strength. The approach using NaOH suggests this may not be successful. 

A less conventional approach may be to partition between two organic solvents e.g. acetonitrile or methanol with heptane or methylcyclohexane. The solubility of TBAP appears good enough in alcohols to dissolve well in that phase, while your amide seems more likely to have less polar solvent solubility. Your methanol-MeCN mix may convolute things,  so maybe try and strip off the methanol (and likely some MeCN will be lost, but you can top up) or if all else fails,  evaporate both solvents and redissolve from scratch.