When playing in the professional division of a PDGA sanctioned tournament, a player can refuse to accept cash in order to maintain amateur status.

In a similar way can a player also refuse cash to avoid going over the $3,000 threshold set to be considered a touring pro?

This would be after they have already accepted cash at multiple tournaments, but once they were close to the $3,000 mark, they stopped accepting cash even though they were winning.

Follow up: what would be the benefits of avoiding touring pro status?

College - they just had a story on the PDGA website during the college championships that a kid did that so he could maintain "AM" status to play for his school.