grammaticality - She had more experience than I, she having [...] or her having [...]

Young's Literal Translation of the Bible contains:

[Luke 15]: 13 `And not many days after, having gathered all together,
the younger son went abroad to a far country, and there he scattered
his substance, living riotously; 14 and he having spent all, there
came a mighty famine on that country, and himself began to be in
want...

But I'd agree that the inclusion of the 'he' is archaic (not 'wrong!') here.

However, I wouldn't quibble about the following example (B) of the usage:

(A) John and Jennie each had more experience than I / me, John (for instance) having produced more than 10 songs already.

...

(B) John and his sister each had more experience than I / me, he having
produced more than 10 songs already.

(Add the 'for instance' in A / B if you wish - it's logical.)