It's a tough situation and an great situation at the same time. It is a statistical rarity to find a program like your son has where the player loves the team and the school and is a significant contributor. Don't take that lightly. Also, there is a ton of overlap across divisions. One of my sons played JC, D2 and NAIA. He now coaches, including as HC in the Valley League where there are players across D1, D2, NAIA, D3 and JC. It is quite common there that the best performers come from all of those divisions that are not D1. So many things can happen when making a transition upward in levels, not the least of which are the previously mentioned class transfer challenges and possible sit time. It is far more likely that a good player get stuck behind a better player or finds himself one slump away from losing his spot in the lineup.
While your son may be a defensive standout at an important position, there will still be emphasis put on the hit tool when trying to get the attention of other schools. And, there will likely be concern that if he is a .313 hitter in D3, he will likely be a much lesser hitter at higher levels where that good pitching is rolled out more consistently. Yes, you can find a weaker D1 or D2 team/conference where that may not be the case but that defeats the purpose. Your son's coach is largely correct in that few D3 position players get drafted. But some of the same applies to weaker D1 and D2 programs. I have seen some REALLY talented position players at D2 get bypassed with any draft interest.
Is it possible that your son transfer to a D1, find success and then find MiLB opportunities? Yes, but at this point, based on what you are telling us, the odds are wildly stacked against him and the chances of things backfiring from a college experience standpoint are quite high.
I'm very much with others who suggest he tries to find an avenue to one of the top collegiate summer leagues to show his skill set. It will scratch his itch to play along side players at the higher levels, give him a chance to be seen and not ruin the good thing he has at his current school.
BTW, there are some here who can help steer you in the right direction with good summer college leagues, if not this summer, then next.