Overview:
•Find a lender that will do "manual underwriting".
•A HELOC may be a good option. (first mentioned in this answer)
•Due to the risk involved, a fixed income person shouldn't build.

Manual underwriting

Manual underwriting means looking at the whole financial picture of the borrower. Is the borrower like your senior - a person without a credit history because they do not borrow money? Or is the borrower a person who has always rented and regularly doesn't pay the rent, being kicked out of place after place.
Both of those examples may not have a credit score in the US. I think you'd agree that one of those people would be a bad choice to lend money to, but the other might be a good choice to lend money to.

Some reasons many entities do not allow their loan employees to do "manual underwriting":

• Financial institutions can mitigate a lot of their risk (of making a bad loan) when they require that the borrower have a good credit score.

• It takes far less training to teach someone to do a loan when the credit score is required to be present and larger than 680, 720, etc. (Less training --> less pay --> less overhead)

• It takes less risk when there is a credit score, because the entity generating the score (FICO/Equifax/Trans Union/Experian/?) is already saying this person pays their bills which is a great start, right?
  Note: it isn't a guarantee that a higher score means you are in better financial shape - as an example a few years ago my credit score dropped 20+ points when I closed a credit card with a high limit. I'm arguably in a better financial position after closing that card because I have less credit now (fewer avenues to get in trouble). There's a good reason for the drop - but it is too far outside the scope of this question, and can be found elsewhere.

• It reduces the chance of fraud when the loan engine can automatically reject loans below a certain score (the lender's employee/agent can't commit fraud by forcing through a loan to a friend/relative by doctoring paperwork, because an internal computer will override and deny the loan)

Many banks will do this if you can talk to the right person.
There are entities that can help you secure a loan with manual underwriting, I have used one of them and I had a great experience (note that I did not need manual) but I don't thing we're allowed to recommend here.

I expect his best bet is a credit union. Many allow manual underwriting.
Some of them are actually non-profit entities and will make 'bad loans' as defined by 'a loan other institutions would not make'. I am not talking about 'bad loan' in sense that the person cannot repay it. I mean bad in the sense that - just for example - if a person had a streak of hard times/bad luck the past few years (job loss or supporting a spouse/parent in bad health caused the person to pay for treatments, instead of paying their own bills on time for example) but that situation no longer applies, and they've otherwise shown a history of being financially responsible.
Interestingly, a number of credit unions experienced less than half of the average default rate for home loans in the 2009 timeframe (2009 was a very bad time period for defaults).

Credit unions have criteria for membership, and if that isn't possible I expect a HELOC would be his best bet for a loan.

HELOC

HELOC == Home Equity Line Of Credit
This is a loan against the current property's value. The amount available will generally be available as a percentage of less than 100%.
If the paid for home appraises for $200k and the institution allows 80% Loan To Value (LTV), then he would have up to $160k (80% of 200k) that could be borrowed to build the new home.

Note that a HELOC may also require a certain credit score to get a good rate, which is why I listed it second.
Perhaps, with no credit score, some entities that would not offer a home construction loan may offer a HELOC but maybe they would adjust to 60% LTV and loan at a higher interest rate. He will have to shop.

Note that he will probably have to make payments on the HELOC while the current home is being built. If you can't make the payments you risk losing both properties.
Historically, there were loan types in the US that allowed no payments but had a 'balloon payment' at the end of the loan - these have largely disappeared due to US Federal regulations put in place after the 2009 financial crisis.

Alternative

Consider selling the home he is in, storing most of his stuff, and living in an apartment while the home is built, or while looking for a new home that may come on the market and meet his needs.

Conclusion

Financially -
He should investigate both a HELOC and any available credit union options. I expect a straight loan with both properties as collateral would be the lowest interest rate, but research needs to be done for his specific case.
Even though you didn't ask, I think he should buy instead of build (see the next point).

Building a home -
I've heard so many stories about a person failing when trying to be their own general contractor, that I do not recommend this option. I've also heard horror stories about general contractors not finishing and going bankrupt.
We are talking about a senior on a fixed income with fixed assets - this seems too risky to me.
I saw a High School friend of mine do this. He had a home that was 'preconstructed' (or some such term) and then assembled on land he inherited (given before his death as his inheritance, but I digress).
He had a fixed price for the two-story home itself, which sounded great.
However, the water and sewage systems went way over budget, and he was on the hook for overruns on those. Power cost thousands to run instead of the few hundred he was quoted from the power company. (Maybe his fault for not getting it in writing)
None of those are optional - so he ended up needing 110% of the value of the home to buy it. Unfortunately, the fixed cost 'preconstructed' house was, to the bank, more like a mobile on a foundation, than it was like a traditional house and so the interest rate was higher.
My friend was in his 30's and could recover through bankruptcy.
(But the the 'family land' he was deeded is now occupied by a non-family member; he was regularly reminded of that until his parents passed.)
Someone who is 65+ making mistakes like those may never recover, transitioning from financially independent, to financially dependent.

Selling a home -

Current house can be expected to sell quickly in this market.

Right... but remember it has to still be good time to sell when it is time to sell his home - and that situation can change quickly. (I know that normally it doesn't change quickly... but it did exactly that in 2009. Financial institutions remember that fact.)

Additional Steps -
It sounds like your senior should be told about Dave Ramsey and his crew - they are big on 'no debt'. Ramsey Solutions provides free financial advice through their website, podcasts, radio show, and other outlets. This article explains the mortgage underwriting process providing additional information. Ramsey solutions does have advertisers, but nothing is required to get the information - not even an email address.