Working in the USA for living expenses only; allowed on VWP?

As a technical matter, your niece as a UK citizen can enter the U.S. under the Visa Waiver Program (VWP); the ESTA is an authorization for the bearer to attempt to enter under the VWP. VWP visitors are only permitted to engage in activities that would be permitted under a B1 (business) or B2 (tourist) visitor visa, examples of which are enumerated on the State Department VWP page:

Business:

• consult with business associates
• attend a scientific, educational, professional, or business convention or conference
• attend short-term training (you may not be paid by any source in the United States with the exception of expenses incidental to your stay)
• negotiate a contract

Tourism:

• tourism
• vacation (holiday)
• visit with friends or relatives
• medical treatment
• participation in social events hosted by fraternal, social, or service organizations
• participation by amateurs in musical, sports, or similar events or contests, if not being paid for participating
• enrollment in a short recreational course of study, not for credit toward a degree (for example, a two-day cooking class while on vacation)

The same page also provides examples of activities that are not permitted, which include

• study, for credit
• employment
• work as foreign press, radio, film, journalists, or other information media
• permanent residence in the United States

"Business" is allowed as noted, but "business" is not the same as "work," and you mention "work." For VWP and B1/B2 visitors, employment, even without pay, is explicitly disallowed. Even volunteer work is only permitted if the work is not normally done in expectation of pay, which would not the case with, say, screenwriting or videography. Interns, trainees, and the like require a J-1 (exchange student/scholar/worker) visa, and there are specific eligibility criteria to meet for each category. Others can be admitted on an H-3 (non-immigrant trainee) visa, though I do not think your niece would readily qualify for one of those, either.

So at best, your niece might characterize this as "short-term training," with incidentals covered, although without additional details of what her day-to-day activities would be, it is hard to say whether they would actually qualify. It would be one thing if this were with a company or non-profit organization with formal guidelines and expectations that could be evaluated. This sounds like this is a personal invitation, which is another matter.

That brings up another red flag, or several: is the videographer interested in your niece's talents, or in your niece? Now, I am sure she, and you, are not naive, and I know nothing about the exact nature of their relationship—he might have the purest of artistic motivations, or they might share an interest in each other (whether creative or intellectual or romantic) undergirding the plan. But to state it for the record, your niece would be on the short end of a tremendous imbalance of power. She would be relying on what sounds like essentially a stranger not just for training and career advancement, but for housing and other necessities as well, in a foreign country, and in an industry notorious for preying upon young women in the city it dominates.

So, while I am not an immigration lawyer and cannot tell you definitively whether her planned sojourn would be legal with the detail you have provided, I am someone who would discourage my own niece from embarking on such a plan unless she pays her own way and makes her own arrangements.

He has been told this by an immigration lawyer!

No, he hasn't. He's lying through his teeth, because as the top answer points out, the law makes it clear that ANY work, regardless of whether or not it's for a regular salary, is not permitted under the terms of her visa. There is no immigration lawyer in the country that would have told him this because it's straight-up not true.

You might wonder then, why would he lie? And I genuinely don't want to sound alarmist, but keep in mind that he would have 100% of the power over her and her freedom while in the US. Considering that she would be violating the terms of her visa while staying there, that gives him ample opportunity to blackmail her into doing whatever he wants to her by threatening to report her to authorities / cut off her funding. Absolute worst-case scenario he could be planning to sell her as a sex slave; LA is a major hub for human traffickers.

Bottom line: he has already shown that he will lie right to your face about serious legal matters just to get her to come to LA. That should be an enormous red flag about his true intentions for her and she needs to seriously consider the danger she would be putting herself in if she chose to go.