How can I get insurance to pay for a memory test?

Dad should be on Medicare A&B so testing should be covered at least 80% and his suppliment paying the rest. If on straight Medicare you should not need a referral to take Dad to Neurologist. Medicare Advantages you may need a referral from his PCP. The Neurologist will order any testing, like an MRI, that he can't do in his office. Labs should rule out any physical problem that may cause Dementia like symptoms like low Potassium levels or a UTI.

Your Dad's MD will schedule this after you speak with your Dad. Before making appointments for this make certain your father has done any paperwork needed to be done with a diagnosis of competency, for instance a POA or DPOA for you to act in his stead when/if needed, and a will or Trust.
Now go to the regular MD with Dad and get any of that paperwork done for instance a Living Will or Advance Directive. Then ask MD for referral and be certain the neuro-psyc doctor is in network of your Dad's insurance, and that any subsequent tests are approved (MRI and etc.)
Wishing you good luck. Discuss all this with your Dad. You and he can "play with" some "fun" tests before going that may tell you a lot; google them. Some of the simple and well know standbys are "Can you draw me a clock set at 2:15" or "I am going to say three words for you to remember for me. Repeat them back to me. Then we will talk about other things for a while and I will ask you to repeat the three words for me".
Be honest about your concerns, that it is important for his safety to follow up, to get "baseline testing", and that this happens to us all, but important for safety and security that things are followed and discussed. Assure him you will be with him all the way, helping him, and never abandoning him no matter what future holds for you both.

His doctor, a neurologist preferably (JMHO) would examine him and determine what kind of test(s) to perform.   

In the meantime....do you live with him or does he live alone?   I'd consider everything that affects him in his life, and think of alternatives.  I.e.,  if meal prep becomes difficult, ordering Meals on Wheels would give him at least one prepared meal daily, as well as extras for holidays.

I'm a firm believer in emergency contact support for anyone, old or young.   Acquaint yourself with the life alert devices, activated with downward movement suggesting a fall and immediate response by the support team of the provider.   A lockbox for his home so emergency personnel have access if he falls or needs assistance is also mandatory IMHO.   

If memory issues affect his ability to manage his finances, step in and help him with bill paying and other financial issues.   

If he hasn't prepared his end of life documents, help find a good, reliable, honest and qualified elder law or estate planning attorney to help him.

These are just for starters.  Others here will offer very helpful suggestions, that you could use to make a checklist for assisting your father.

You're wise to address this now.  Best wishes and feel free to post back if you have any questions on my limited recommendations.