My ice auger makes a 6" hole (I think). I could try to compare methods this winter but whatever lid I use needs to fit through the hole. I fear a 6" foot print is too small. The alternative is that I mark each location with my GPS, use the sonar/pole method this winter, then measure again with the lid/pole method once the ice is off.
Around here, when spearing northerns through the ice, we drill holes so that they just barely connect. You could drill three holes in a triangle so that their edges just barely meet. This would give you more room to work with.