I'm not an attorney, and have limited knowledge, but it is my understanding that Lenders can do what you are requesting, but it will depend on whether it was set up as a Revocable or Irrevocable Trust and who the Trustee is now.  

You're basically putting down 5.25% + having the irrevocable pmi for life of this loan.

It is owned by an irrevocable trust where I am the trustee.

This way, my corporations never show a profit and I avoid double taxation.If a person has a lot of equity to protect against lawsuits I went with an Irrevocable Living Trust.

One is to move your equity/ownership in LLC into an irrevocable trust where you have the control to manage the LLC as the manager, but in the capacity as trustee of the Irrevocable trust.

Charitable trusts require that that you give up legal control of your property, and charitable trusts are irrevocable -- once the trust becomes operational, you cannot change your mind and regain legal control of the trust property.How It WorksThe most common type of charitable trust is called a charitable remainder trust.

All contracts, options, or other devices not based upon a substantial consideration, or that are otherwise employed to permit an unlicensed person to sell, lease, or let real estate, the beneficial title to which has not, in good faith, passed to such party for a substantial consideration, are hereby declared void and ineffective in all cases, suits, or proceedings had or taken under this chapter; however, this section shall not apply to irrevocable gifts, to unconditional contracts to purchase, or to options based upon a substantial consideration actually paid and not subject to any agreements to return or right of return reserved.History.

If you haven't already, something you might look into is an irrevocable trust. 

a lot of this depends on how much TRUE cash or equity your trying to protect.. if your just putting minimum down on turn key rentals.. there is not a lot of equity to protect and would be rare to the extreme anyone would pay to go the full monty getting a judgement against a house that by the time they foreclose and pay off our secured lenders and real estate fee's etc there is not much left.keep in mind 20% equity in the Average C class rental is really no equity.. nothing really to get.if your worried about cash for your daughters you can set up an irrevocable trust for them right now. 

When managing a trust for someone else, then it will be revocable or irrevocable depending on how it is set up.