I strongly recommend that you speak with a solid Catholic confessor before going through with this. If possible, it would be best to also discuss the situation with a Catholic elder-care attorney who is knowledgeable about his faith to get a clear understanding of the law before you speak with the priest.
At first glance, it would seem to be deceitful to persuade an institution to accept you when you know it would refuse you if it knew your true intentions. However, the way you describe the situation, it seems that the law permits people to do this.
The way you describe it, you would be making a true statement to the institution, “I am not now on Medi-Cal,” and whether or not you switch to Medi-Cal in a month is something the state does not permit the institution to do anything about. So, there is some possibility that the action might not be sinful.
But I wonder if it is that simple. It may be that the institution can take simply limit the number of beds that are available for Medi-Cal patients. Then they can tell you, “Oh, we will put you on the waiting list of 235 people waiting for our 15 beds.” (I have no idea, I am simply analogizing to the situation in my own state, where you could not do something like this so easily.) Or, it may be that the question they ask you up front is one that you could not honestly answer at all, like “I affirm that I have no intention of using Medi-Cal during the next three years.”
God bless you in your decision.