Do any banks offer fractional joint-ownership mortgages?

No, big banks will not do this - they are for the most part "mortgage factories" that issue simple, conforming mortgages and then sell them to consolidation companies that then issue "bonds" to investors that get paid when the mortgages are paid. Any bespoke mortgages would likely have to stay on their books, so they are less likely to do them.

A local bank might do it, but it's not a simple thing. What happens if the child decided not to pay the mortgage? Does the bank then get to foreclose and sell the house even the trust owns 33%? Does the trust then get 33% of the proceeds from the bank selling the house? If the house is underwater, does the trust have to partly compensate the bank? Is that fair to the trust?

It's possible but definitely not simple. A rent-to-own arrangement with the trust may be simpler, where the child pays rent to the trust and builds up equity in the house over time. You just need to work out what a reasonable interest rate is to calculate the payment and track the interest and principal amounts accurately.

D Stanley gives an excellent and correct answer, and I believe that in your case a "rent to own" approach is probably going to be best. However I wanted to add a more general answer.

Banks do provide mortgages to people who want to fractionally own a house. However they always like to do it by providing a single mortgage to which everybody with a share in the house is party. Specifically they want to make everybody "jointly and severally" responsible for the mortgage - i.e. everybody is responsible for all of the mortgage if somebody stops paying their share. I've been part of one of these mortgages, when I bought a house with two unrelated friends.

In these cases the fractional share of the house, and the statement of how much of the mortgage each person pays, is written up in a contract entirely separate from the mortgage. This is to get round the the problems detailed in the other answer. If one person 'walks out' on the mortgage then the other parties either have to pay that person's share (and possibly pursue them legally for breach of the sharing contract) or allow the house to be sold.