I DON'T think the White Lady would be homophobic, Ze'mer was gay and the White Lady specifically calls her beloved. Ze'mer still calls the the White Lady 'Our Queen', and Ze'mer specifically cites the Mantises as having denied their union (due to xenophobia though not homophobia).
Amongst its hateful kin that did deny our union, that did reject che's… outside-ness.
So, canonically evidence points to the White Lady not being homophobic! If Hornet brought Shakra home, I think the White Lady would have nothing but kind things to say and be delighted for her.
Buuttttt, I actually do think the White Lady might take issue with SPECIFICALLY Lace. I would give a lot of things to see the two of them forced to interact for a long period.
Why? Well, in case you've forgotten, in Hollow Knight, the White Lady thinks the vessels are empty still. She's not overtly cruel, but she directly requests that the Knight take over as the new Hollow Knight. Like, even though she has no idea what compelled them to seek her out. And she specifically acts like the Knight agreed and tells them not to waste time. By every indication we have, she seems to think the Knight empty.
Further, the White Lady by her own admission already feels somewhat guilty about what she did, and that's while believing that the vessels were empty creatures without emotion/thought/etc.
And.... Lace I think might actually be really difficult for her to process emotionally. Lace is a construct. She is a bug made entirely of soul. She's not possessed by the void (anymore) but she is still a pretty close parallel to the Vessels, very intentionally. Lace herself describes herself as an "empty shell", "not life", and "a husk".
Lace is not a bug. She's a creation of a pale higher being meant to serve as their tool. And Lace is also profoundly and undeniably a person. She has a voice, she has thoughts, and feelings, and emotions. The combination of Lace being so close to the Vessels and also completely and utterly a person could be really hard for the White Lady to swallow.
Further, even though I think Hornet is pretty clear in game that she didn't consider her siblings empty, any world where Lace and the White Lady meet would be post Sister of the Void. At which point, assuming Hornet bothers to explain what happened, the White Lady would have to very directly confront the fact that none of the Vessels were empty. It's not just the Knight who shows up to save Hornet. We see the siblings acting as a collective, in an explicit act of care towards their sister. Lace was part of all of this. She was there.
All of that together might be a bit much for her to take. I hardly think she'd be violent or overtly cruel, but I could easily see her slipping into a place of denial and refusal to engage with the concept that the children she discarded were in fact, children. And, as an extension of that denial, Lace cannot actually be a real person. She's a trick. A very clever creation of deception. A husk shaped to act like a child. Even if she pretends to go along with it for Hornet's sake, I could see her holding onto that belief for quite a long time.
How exactly this would play out, I don't know, but I could see it as entirely possible and a very interesting dynamic to explore. She could be an excellent toxic passive aggressive force in Lace's life, getting in the way of her relationship with Hornet while setting off every alarm in Lace's body. It'd be fun if nothing else.