Understanding "que je ne lui avais pas encore vu mettre" in La porte étroite

Question

The question is on lui as occurring in this passage from La porte étroite by Andre Gide.

Dès le vestibule, j’aperçus Alissa ; elle semblait m’attendre et vint aussitôt vers moi. Elle portait au cou, dans l’échancrure de son corsage clair, une ancienne petite croix d’améthyste que je lui avais donnée en souvenir de ma mère, mais que je ne lui avais pas encore vu mettre. Ses traits étaient tirés et l’expression douloureuse de son visage me fit mal.

1. Can we say la in place of lui without making the sentence ungrammatical?

2. If yes to 1, does lui still read much better (more natural) than la?

Background

It seems to me that lui in the sentence can be assimilated to designation of agency in a sentence such as:

Je fais laver la voiture à David.

But at least in English we are not restricted to:

I never saw it worn by her,

but can also say:

I never saw her wear it.

In French also, at least when there is no second item (as here, the cross) to confuse one, we seem to have:

Je la vois partir.

Answer 1

Sometimes, lui is the only option and is used for both feminine and masculine. Ex :

J'ai vu ta sœur hier, je lui ai dit bonjour.

In the sentence you gave (Je la vois partir), the masculine form doesn't use lui, but le :

Je le vois partir.

When lui can be feminine, it becomes elle, not la : "Je vais avec lui/avec elle". But when it's in a position such as "Donne-lui sa bague", it's invariable.

So to answer 1., no you can't say la instead of lui.