Can a patient be transferred from one nursing home to another for better care?

If your brother is of sound mind and can make his own decisions clearly, then he can decide to go elsewhere. If he is cognitively impaired, his PoA or guardian can make the decision.

We had the same problem with my MIL: once she got into the LTCF she never got out of bed. Eventually we moved her to a different place and she has been coaxed out of bed to eat meals, be social and even go on field trips. It is blowing us away as we thought she'd never get out of bed (even though she was perfectly healthy). But now her muscles are mostly gone (she was in bed for 2 years) so she's in a wheelchair, but at least participating.

Her original LTCF was a good place but very under-funded and under-staffed (it was run by a local church next door to it). Now she's again in a faith-based facility but a much larger one with far more experience and staffing. They have worked magic. FYI my MIL is on Medicaid. We feel so blessed at the care she is receiving.

Yes.

Sure, you need to call on other Nursing Homes in your area and apply to have them admit him. Scout out the ones you like and apply. There may be a waiting list though.