Get Armor-stand part inside GUI

So I have created a GUI in which I have rendered an Armor-stand. My goal is it to get the exact part that the player clicked at (an arm, a leg, the torso, e.t.c.).

I can't use Minecraft#objectMouseHover, since the Entity hasn't been added to the world (it's only rendered). For the same reason, I can't use World#rayTrace().

Is there an easy way of accomplishing what I want to do, or do I manually have to calculate the "inverse projection"?

Here's the code that I use to render the Armor-stand (basically what Minecraft does inside the player inventory):

    private void renderInside() {
        final int i = guiLeft + windowWidth / 2;
        final int j = guiTop + windowHeight / 2 + 80;
        final int scale = 80;
        RenderSystem.pushMatrix();
        RenderSystem.translatef(i, j, 1050.0F);
        RenderSystem.scalef(1.0F, 1.0F, -1.0F);
        MatrixStack matrixstack = new MatrixStack();
        matrixstack.func_227861_a_(0.0D, 0.0D, 1000.0D);
        matrixstack.func_227862_a_((float)scale, (float)scale, (float)scale);
        // horizontal scale
        Quaternion quaternion = Vector3f.field_229183_f_.func_229187_a_(180.0F);
        // vertical scale
        Quaternion quaternion1 = Vector3f.field_229179_b_.func_229187_a_(-previousPitch * 20.0F);
        quaternion.multiply(quaternion1);
        matrixstack.func_227863_a_(quaternion);
        // renderYawOffset: everything but base plate
        float f2 = armorStandEntity.renderYawOffset;
        float f4 = armorStandEntity.rotationPitch;
        float f5 = armorStandEntity.prevRotationYawHead;
        float f6 = armorStandEntity.rotationYawHead;
        armorStandEntity.renderYawOffset = 180.0F + previousYaw * 20.0F;
        armorStandEntity.rotationPitch = - previousPitch;
        armorStandEntity.rotationYawHead = armorStandEntity.rotationYaw;
        armorStandEntity.prevRotationYawHead = armorStandEntity.rotationYaw;
        EntityRendererManager entityrenderermanager = minecraft.getRenderManager();
        quaternion1.conjugate();
        entityrenderermanager.func_229089_a_(quaternion1);
        entityrenderermanager.setRenderShadow(false);
        IRenderTypeBuffer.Impl irendertypebuffer$impl = Minecraft.getInstance().func_228019_au_().func_228487_b_();
        entityrenderermanager.func_229084_a_(armorStandEntity, 0.0D, 0.0D, 0.0D, 0.0F, 1.0F, matrixstack, irendertypebuffer$impl, 15728880);
        irendertypebuffer$impl.func_228461_a_();
        entityrenderermanager.setRenderShadow(true);
        armorStandEntity.renderYawOffset = f2;
        armorStandEntity.rotationPitch = f4;
        armorStandEntity.prevRotationYawHead = f5;
        armorStandEntity.rotationYawHead = f6;
        RenderSystem.popMatrix();
    }