How to best equalize tangible items to kids before Mom goes on Medical Assistance (MA)?

You are speaking of selling a portion of your Mom's estate now so I am assuming she is living, but cannot make these decisions, and that you are her POA. You should sell this car for fair market value. It is your fiduciary duty to do so. You are not there to give favors or dole out portions of things belonging to your Mom, but to sell them (if your POA gives you that right) for fair market value and put the funds into Mom's estate. That takes all burden and onus off of you to do otherwise.
Your mother can keep her home and a vehicle and still collect medicaid in many states. Medicaid is a joint state and federal government program so rules vary state to state. If you have already applied for medicaid then you will have already filled out the forms with ALL assets that belong to your Mom. After her death there will be claw back, and the executor, whomever is assigned in the will, will again have to sell these things for fair market value, almost certainlly.
Sure wish you the best. Check with your authorities in your state.

With the ‘promised’ items of no great value, here is a method I’ve been involved in twice: Each person gets a ‘pick’ in turn. The cash value is agreed between everyone if they are not just ‘memory’ value, or if there are strong bids for the same item. There’s an agreement about whether ‘memory’ trumps someone who just wants to sell the item for profit. Someone makes a list of who takes what, and the value in each person’s column. At the end of the session, the value for each person is added up. If Medicaid is involved, the totals go into the estate. If it's just between the family, it's equalised with cash.

This works well if the people involved are well behaved. It stops one person getting all the ‘valuables’, and gives an equalisation method. For example, my ex wanted most of our furniture when we split, and I needed more cash. My sisters and I did it slowly with M’s things, and enjoyed talking about our memories as we did it. We know what’s now where, and it is another conversation starter when we visit and see things again.

IMO you should make an appointment with a Medicaid caseworker. Face to face or Zoom.

You need to be careful what you do when dealing with Medicaid. I was told that any Will you may consider null and void. Why, because all Moms assets must go towards her care. They need to be listed on the application but a home and car are exempt assets until death when Medicaid recovery gets involved. I would wait to sell that care asking the caseworker what do you use, dealer trade in or private sale. Then what condition is the car in, fair, good or excellent. Ask if getting a dealer estimate would be good? Once you sell it, you have to include the proceeds in the spend down amount. My Moms house was already up for sale so I don't know if it had to be. By not selling it, someone will need to be paying the taxes p, utilities and upkeep.

Medicaid usually only allows a family Caregiver who has lived in the house and taken care of the LO for 2 yrs to remain. A disabled child can remain. But with both, the person has to be able to pay taxes, pay utilities and pay for upkeep. The house cannot be rented without Medicaid knowing. It could be considered income and effect the income cap.

If Mom only has a couple of months of money you better get that application started. In my State you have 90days after applying to spend down assets and get info needed. If Mom is moving out of the AL, you need to know where she will be placed. If this is not done in 90days, you start all over.

My MIL was told when she had her Will drawn up, if she found after the fact there was something she wanted to leave someone, she needed to have a codicil. I know she asked me what I wanted so I assumed she did it. She didn't. We really didn't care, but you should have seen the neighbors who came over making claims. One was her car. I think she told all the grands they could have her car when she died. POAs daughter got it. Was it in writing, husband never asked.

When it comes to the contents of the house. Its been 5 yrs but I don't think the caseworker ever asked that question. Mom did not have anything of value. I gave most of it away and my brother came and got some furniture Mom had. If there are antiques that are worth money, they need to be sold if worth a lot. The car needs to be sold at fair market value. It matters if a car sells for 900 or 9000. It means thats money for Moms care and Medicaid wants you to get what that car and house are worth before they pay anything out for her care. You need to talk to a caseworker, which is free, or an elder lawyer who is not before you do anything with your Moms belongings. If this is not done right, Mom could be penalized. Then you and siblings may either be taking Mom in or paying for her care until the penalty period is done. At this point do not spend any money that is not used directly for Mom. No gifts, no furniture for a sibling.

Don't try to divide things up by, Johnnie wants Moms $500 worth of silver, Jane wants the China thats worth $1000 so what does Johnnie get thats worth another $500. You will drive yourself crazy. Everyone should take what they want regardless of what its worth.

You don't base your asking price using KBB's dealer price. That's only what to expect if you sell to a dealer.

You have to use the private party price which, unfortunately, is the higher of the two.