I wonder how much consideration the founders put in to the effects of non-magical chemical interventions.

On the day before the sorting I could conceive and then carry out the following plan:

• Pack a lunch packed lunch which includes a beverage dosed with MDMA.
• Ensure that I will eat this food in preference to purchasing off the trolley. Either by writing a note or by giving my parents instructions to remind me quite seriously.
• Drug myself with an amnesiatic cocktail and so forget the plan.

MDMA would likely be sufficient to influence the sorting. Especially if combined with extensive psychotherapy over several months. Since you have allowed influence by the parents there is even more scope for influencing the sorting by non magical means. Chemical and psychological interventions can make a huge and somewhat reversible influence on psychological traits.

There may be similar non magical ways to enhance a polyjuice based plan. Legal name changes. Chemically enhanced hazing to convince the volunteer that their actual name is wedrifid, etc.

Wizards are notoriously narrow minded when considering non-magical loopholes, especially in the MoR!reality. Bypassing the hat as an eleven year old may be difficult without assistance but should definitely be possible with parental assistance. As an extreme measure:

• At age 10 backup all your memories.
• Have the parents introduce retrogade amnesia and solid brainwashing over months or a year.
• Restore the memories at the Christmas holidays and work hard to reverse all changes. This should be possible because magical intervention can be used in the healing process without the hat interfering. Apart from healing spells things up to and including self cast imperius can be used.