Given how serious her loneliness issues are, seducing Mami is probably a lot easier than it sounds.
If it was no one but Mami, it would be easy. The difficulty is in developing the relationship naturally, in conjunction with maintaining the relationships with Kyouko and Sayaka. That means we have to be gentle and circumspect about things, at least at the start.
Sayaka has referred to Kyouko as Homura's "Waifu", and Kyouko has referred to Sayaka as Homura's sidekick, but with Homura correcting her to note that Sayaka is "more than a sidekick". Homura has very strong feelings for Kyouko, though Kyouko's feelings for Homura seem more protective (or at least she's sublimated the moments when Homura was coming onto her pretty strongly). Homura has been surprised to find that she considers Sayaka a friend (but that's definitely growing), but Sayaka herself has been not-so-subtly flirting with Homura.
So Sayaka > Homura > Kyouko in terms of more romantic feelings, and Kyouko > Homura > Sayaka in terms of more friend-like feelings (but growing in various ways). Sayaka and Kyouko seem to have gotten into a fairly reasonable rivalry-friendship.
Now, what happens if Homura suddenly pulls Mami into that mix? Mami has a tendency towards extreme attachment, and if Homura pulled a Seductive Englishwoman on her, Mami would glom onto her without regard for the other relationships.
Sayaka and Kyouko could both roll with it because the friendship connections are strong enough, and the romantic connections are still tentative at best, but it would be... disappointing, and upsetting. So if Homura wants to allow a connection to Mami to grow, she has to make sure Mami is aware of these undercurrents, so as not to just crash through them.
Sayaka is likely to be at the high end of touchy-feely during dinner, and even more so if discussions afterwards move into territory Homura is uncomfortable with. Kyouko, meanwhile, uses a regular nickname for Homura, something she doesn't even do with Mami. Of course that's related to the fact that her feelings for Mami are rather more complex, but it's still indicative of how Kyouko feels about Homura, and just one of many things that will be running through Mami's mind.
I expect Mami to have a reasonably high probability of at some point tonight confronting Homura over this. And Homura is going to have to own it, being honest with herself and the others. That event is going to control what options there are for connecting to Mami.
It's unfortunate that we don't have enough sanity to safely unlock Mami's libraries, but it's probably just as well. Let things settle on the current connections, before trying to expand them.
