Does life insurance payout disqualify widow from Medicaid?

You are right. No transfers. The money belongs and should be used by your dad.
The insurance can be used to pay for his care.
That’s a good thing right? He needs care and can now continue to pay his own way.
Perhaps there are more services available at his AL that would help keep him in better health longer.
When he has used up his excess he can file for Medicaid.
Are you his POA? You could help him prepay his funeral if that hasn’t been done. A monument for his spouse and/or himself would be another item he could pay for.
He could get health care that isn’t covered by Medicaid like dental appointments, new glasses, a nice rolater if he needs one. Perhaps a new recliner. Perhaps update his wardrobe. As long as the money is spent on him or for his care is the idea.
He should see a certified elder attorney who has experience with Medicaid laws for his state. He can not do anything that would be considered gifting the funds.

In my state Medicaid whether for insurance or paying for care in a NH, pays for eye care and dental. But these things can be paid for prior to being put on Medicaid with no problem.

First: IANAL and always seek an elder law attorney (preferably with estate planning experience) for these kinds of things. SSI and medicaid laws are complicated, full of gotchas, are updated several times per year, and vary by state. ..You simply can't get accurate advice online.

The first thing you should do is spend some of that money towards such an attorney.

That said:

If he's on SSI/medicaid for disability and under 65 he may be able to setup a self-settled special needs trust, which will allow the money to be used to improve his life while not affecting his benefits. I would pursue this option first and foremost.  But know that they can cost in the ballpark of $5k to setup - and you _must_ have a lawyer knowledgeable in SNTs to set it up, and the trustee needs to be careful how it's spent (I actually paid an additional $1k to my attorney so that I would be able to get advice as needed without incurring an hourly fee).  Anything else will just result in massive headaches in the future... 

Also, not all transfers are bad. Cashing the check wont instantly disqualify him or something.  The month he receives the check may count as unearned income, reducing his benefits to $0 for the following month. In some states this could also mean suspension of medicaid as well. But in both cases it's possible to avoid that via a hardship waiver, if 1 month without benefits would be unacceptable.  

What he has on the 1st of the month is more important. Ideally you cash the check early in the month and use it on exempt resources before the next month. He can also suspend benefits for some amount of time (12 months for SSI, varies for medicaid) and have it reinstated once he's below the resource limit. (After the time limit he may have to reapply like he did initially, but there's some options there. depends on his situation.)   This is called spend-down.

https://www.specialneedsalliance.org/the-voice/utilizing-the-spend-down-option-to-maintain-ssi-andor-medicaid-eligibility-2/ gives a decent overview.

Thing not to do:

1. don't: hide that he received it, or lie about the amount, or try to buy something above fair market value, etc.

That can result in a fraud of similar fault determination - for him AND you.

Note that failing to report it can be perceived as trying to hide it, so make sure it's reported when he receives the check. While there's a time limit on reporting don't stress too much about it, as "good cause" reasons are accepted - e.g. waiting for advice from the attorney. You just want to avoid the scenario where they find out some months or years later and start asking questions.

2. don't: gift the money or try to disclaim it.

This is will result in a period of ineligibility, based on the value of the gift.

Depending on his situation a short period of ineligibility may be acceptable. If he's considering such an option I would work with the attorney to figure out how much he can afford to gift.

3. don't: hire a friend or family member for services.

He _can_ if really needed, but before services are provided you'll need to get a notarized contract showing the exact costs, which must reflect fair market value. I'd still say avoid this option if possible, though.

4. don't: buy things for the value.

He can buy new furniture, electronics, etc. But if it's not something he actually needs/uses they'll count it as an available resource, as if he still has the money in the bank. That said, I would avoid jewelry or watches or really anything that's good at retaining value.

Good luck, and sorry about your loss.

He is over 65, not yet ready for nursing care, lives on his own, but has compromised health and will probably need assisted care in the future (who knows when). The life insurance payout isn't huge and the amount he wants to gift isn't huge. In my research, I have found that the penalty for gifts made is determined by the "state divisor rate" per month. So, if the divisor rate happens to be $5K and a $10K gift was made, there is a two month penalty. Doesn't seem that bad? Right? What am I missing here?

There's two rates you have to consider - SSI and medicaid.

SSI is amount / 750 (currently) months, while medicaid (amount) / (state penalty divisor) days. Since the state divisor is based on the (very high) cost of nursing homes the penalty period tends to be short, while the SSI benefit not so much.

Also, in some states you have to be receiving SSI to receive medicaid. So the SSI penalty period can create problems with medicaid as well. It's a muddy topic.

See if there's an elder law help line in your state. You may be able to a quick answer for free or cheap.

Is your dad on SSI?

Any money that is gifted will create a penalty period.

For Medicaid purposes he can spend down the money on anything for himself. A new wardrobe, a lovely vacation, a nice cruise, eating in fancy restaurants every night, or, if he needs a care center before he has spent it all, self-paying for it until it is time to apply for Medicaid. Some Assisted Living places do not accept Medicaid until the resident has been self-paying for a certain length of time. If the places you like have requirements like that, then self-paying for that period of time might be a good use of the money.

The penalty period may not be long. But how is it going to be paid? Let's say the penalty period is 2 months, and the monthly rate is $6,000. Where is that $12,000 going to come from? Remember at this point Dad only has $2,000 in assets.

In any case, you and your dad should definitely consult a lawyer specializing in Elder Law.

So...

Is this coming from the perspective that your father has received this windfall, he wants to do something nice with it, he likes the idea of presenting a goodish lump sum to - you? Grandchildren? Anyway... - his offspring, but you and he are rightly being circumspect about how this might affect future Medicaid applications?

It just doesn't seem worth taking a chance on, really. Couldn't he do something else for you all?

Timb, where does Dad live now? Your profile says Assisted Living. Is that what you mean by "lives on his own"? Just a little confused ...