Mid Atlantic Dad, Have you looked at a league like the New York Collegiate league. It is a step up from most local leagues, but plays plenty of D1, D2 and D3 players.
Also sometimes it differs from team to team in a league. My son played for the Settlers in the Great Lakes league, He was a left handed D3 pitcher at the time, but there were other d3 players on the team getting playing time. They had an OF from Elmhurst, David Wolak, and a first baseman from Denison, Ryan Mulligan. Both players got plenty of playing time. The Settlers won the league that year. Other teams in the league had no D3 players what so ever. Some would not even consider using D3 players.
My son also played in the Valley League. He played for Covington. Covington is a team that has players from all levels, and do make the playoffs with some regularity. But other teams in the league may not look at a D3 position player.
Sometimes you need to go beyond the league and look at the teams in the league.
Some of it is also who your head coach has connections with. When my son was at OWU, the coach was still reasonably new. He was still building those connections, so he was very careful who he sent to the upper leagues. It was very important that they do well and be a good citizen. He did not want to send a player that would not help the team. It was important for him to build those relationships. Now he is sending players to the great lakes and prospect league on a regular basis. He also sends players to the NYCBL mentioned above.