Welcome to FT!, Wet Hen

To make a voluntary change in advance, what UA is stating is correct. 2 hours should be a fairly safe and comfortable connection in most cases even with a family.

You can call and escalate to a supervisor but it will be repriced including the SNN segment and likely will be very pricely. This is not going to be done for free.

An option is a Same Day Change or Standby. WIth SDC you can in the 24 hours prior to departure change to another flight (for a fee is not an UA elite). It requires the same fare class availability as your purchase and required there are the number of seats unsold. You can also showup for the earlier flight and go standby but that will almost always be in economy.