#JavaScript (ES6),  79  77 bytes

Takes input as (B)(A), where \$B\$ is a list of sets and \$A\$ is a list of lists. Returns a falsy value (false or undefined) if \$A\preceq B\$, or true otherwise.

B=>g=([a,...A],u)=>a&&B.every((b,i)=>u>>i&1|a.some(v=>!b.has(v))|g(A,u|1<<i))

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JavaScript (ES6),  79  77 bytes

Takes input as (B)(A), where \$B\$ is a list of sets and \$A\$ is a list of lists. Returns a falsy value (false or undefined) if \$A\preceq B\$, or true otherwise.

B=>g=([a,...A],u)=>a&&B.every((b,i)=>u>>i&1|a.some(v=>!b.has(v))|g(A,u|1<<i))

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#JavaScript (ES6), 79 bytes

Takes input as (B)(A), where \$B\$ is a list of sets and \$A\$ is a list of lists. Returns a Boolean value.

B=>g=([a,...A],u)=>!a||B.some((b,i)=>u^(v=u|1<<i)&&a.every(v=>b.has(v))&g(A,v))

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#JavaScript (ES6),  79  77 bytes

Takes input as (B)(A), where \$B\$ is a list of sets and \$A\$ is a list of lists. Returns a falsy value (false or undefined) if \$A\preceq B\$, or true otherwise.

B=>g=([a,...A],u)=>a&&B.every((b,i)=>u>>i&1|a.some(v=>!b.has(v))|g(A,u|1<<i))

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