The Dink Network

Are you only changing Dink's base idle during the cutscene or consistently afterwards?
If not you could always do something like this.

//Get Dink's coordinates
int &getx = sp_x(1, -1);
int &gety = sp_y(1, -1);

//And spawn a fake Dink for the cutscene that you can control instead
sp_nodraw(1, 1);
int &manualdink = create_sprite(&getx, &gety, 0, 436, 1)
sp_pframe(&manualdink, 1);

Then proceed as you have done, and to kill it off

//Kill the fake and redraw the real
sp_nodraw(1, 0);
sp_active(&manualdink, 0);

Edit: thinking further, it might be best if you do it this way considering Dink is consistently Brain 1, whereas you can use a different brain on a fake that you control the frame or sequence for manually.