If the aircraft were to be flown in Canada, the following regulation would apply:
"Unserviceable and Removed Equipment - Aircraft without a Minimum Equipment List
605.10 (1) Where a minimum equipment list has not been approved in respect of the operator of an aircraft, no person shall conduct a take-off in the aircraft with equipment that is not serviceable or that has been removed, where that equipment is required by
(a) the standards of airworthiness that apply to day or night VFR or IFR flight, as applicable;
(b) any equipment list published by the aircraft manufacturer respecting aircraft equipment that is required for the intended flight;
(c) an air operator certificate, a private operator certificate, a special flight operations certificate or a flight training unit operating certificate;
(d) an airworthiness directive; or
(e) these Regulations.
(2) Where a minimum equipment list has not been approved in respect of the operator of an aircraft and the aircraft has equipment, other than the equipment required by subsection (1), that is not serviceable or that has been removed, no person shall conduct a take-off in the aircraft unless
(a) where the unserviceable equipment is not removed from the aircraft, it is isolated or secured so as not to constitute a hazard to any other aircraft system or to any person on board the aircraft;
(b) the appropriate placards are installed as required by the Aircraft Equipment and Maintenance Standards; and
(c) an entry recording the actions referred to in paragraphs (a) and (b) is made in the journey log, as applicable."
Where (1e) refers to "these regulations", the appropriate line would be:
"(j) a means for the flight crew, when seated at the flight controls to determine
(i) the fuel quantity in each main fuel tank,"
So, it would appear that there flight in the aircraft would not be permitted at all, and there is no choice that the pilot must legally make an entry in the log book.
As for structure, Unless the pilot is actually qualified and willing to make a formal declaration that the "structural issue" does not cross any of the lines in the following guidance, he/she would be a fool to fly the plane, because by flying it, the pilot has declared it to be airworthy for the knowledge they have of it, and not in conflict with the standards of airworthiness, which include the following:
(b) Structural Strength
Information Note:
The questions contained in this paragraph shall be applied to alterations of an airframe, engine, propeller, or component.
Does the modification or repair alter:
(1) a principal component of the aircraft structure such as a frame, stringer, rib, spar, skin or rotor blade?
(2) a life-limited part or a structural element that is subject to a damage tolerance assessment or fail-safe evaluation?
(3) the strength or structural stiffness of a pressure vessel?
(4) the mass distribution in a structural element?
Information Note:
This might involve the installation of an item of mass that would necessitate a structural re-evaluation.
(5) a containment or restraint system intended for occupants or the storage of items of mass (e.g. cargo)?
(6) the structure of seats, harnesses, or their means of attachment?
Letting alone your safety, is flying an aircraft with a known defect worth the legal and insurance risk? If it is not legal, it is not insured. The owner who let you fly it with the known defect, would probably expect you to pay for it when his insurance would not.