Most assisted living communities in the area I live, Northern California, do not provide furnished apartments. However, there are several move management services that can assist you in purchasing and/or renting furniture for your mother. If you are interested I can provide you with a detailed list of what you will need (right down to linens and glasses) so that you can get started.
With my move management background I can assure you that you are correct in not moving her furniture cross country, it will cost more to move her furniture than it will to replace it. Another thought to keep in mind is that if you enlist the help of a senior move manager they will also assist you in selling your Mother's current furniture...this can also help to offset the cost.
Feel free to contact me if you have any questions or would like some referrals to services that may be able to assist.