Ok, it seems the Speaker of the House’s relationship to the Sergeant at Arms is more a matter of tradition and in being responsible for the budget of the building which includes budgeting for security.
However, there’s no established rules between them specifically. It’s a traditional relationship and expected – and there have been court rulings firming up that idea – but there’s not rules established for it.
“House rules and manuals reference the SAA assisting the Speaker but do not spell out the exact reporting relationship.”