Nursing home wants to move my father to another facility, opinions needed.

I'm curious what exactly is the reason they feel unable to meet his needs, is he exit seeking?

So what is the reason they are giving you for a move? Do you feel he isn't getting adequate care where he is? If the main problem is that he forgets he shouldn't walk I'm doubtful that a memory care would be any better at preventing falls.

This really sounds like a pretty wishy washy reason to make a move, especially if you chose this place because of it's proximity to you. You can certainly explore if there is a good option near you, but I certainly wouldn't make a difficult change based on that alone. I'm wondering if there is something else they aren't telling you.

The term 'locked down or locked unit' sounds awful, doesn't it? I prefer 'secure unit' which is more accurate. Residents cannot be locked down or restrained in any manner at any time, but they can be kept safe(er) in a secure unit. That said, not all locked units are the same. It may be a that a secure unit will provide a more stimulating environment for your father. A good facility makes multiple adaptations to a wing of the facility in order to better accommodate the needs of residents with dementia/Alzheimer's disease (they are different) and provides specialized dementia training to the staff who will be assisting them. There are unique activities designed for them and residents tend to find it easier to develop meaningful relationships with staff and friendships with one or two similar residents that they see regularly as compared to a large facility with lots of unfamiliar people and rotating staff. An average facility puts magnetic locking doors on the end of a hall and sections off some rooms and calls it a secure unit. Some of them are OK to good, others are poor to poorer. Please visit any facility you are considering beforehand. Looks can be deceiving - it's about care, not the fountain in the courtyard.
On the other hand, your Dad may be more than the facility believes that they can safely provide care for and this is a gentle way to tell you that. Falls happen, unfortunately, and they are difficult to prevent, particularly for residents who have dementia and don't know it. They forget they can't walk and try to go to to the bathroom without assistance, for example.
If you have a good relationship with the facility, talk to them about their concerns for your father and ask for their input on his needs. If a transfer or discharge is determined to be in his best interest, the facility has resources to assist you with Medicaid beds. If the facility you want is not available, they can put you on a waiting list. Until then, you can discuss additional interventions to keep your father safe, content and mentally stimulated.

If another facility doesn't except Medicaid, then he can't transfer. He doesn't have the money to private pay and you r not responsible financially for paying for it.