Having worked with many Eagle projects, paperwork, advancement chairs etc. I think your district adv. chair caved in to those in your troop to stall your son's progress.
I suggest you correspond with the dist. adv. chair, in writing with cc's to the DE, Dist. Commissioner, Council adv. chair and even higher. In that correspondence you ask the dist, adv. chair for verification of her statements about fundraising, start of project or any other potential issue you feel is improper. By including the cc's you put her on notice that she has be answer truthfully. The cc's will understand the potential issues and will not feel blinded sided should you have to approach them directly. I would also include statements such as "I understand you to say xxxx" "I have verbably asked for an explanation and you have only stated it is BSA policy"
This approach may not make friends depending on how she is hooked into the district and council but should get you honest answers. I would expect that the appropriate district and council personnel will bring this to a mutually agreeable conclusion. As noted by others, this is a adult interference issue.
