With the caveat that I think most Britpop is bad and I don't really understand how the genre is supposed to be defined, I'll nominate
The Boo Radleys - Giant Steps (a borderline shoegaze album, but it's hooky enough to count as Britpop I think. Wake Up! probably fits a little more stylistically but isn't as good)
The Auteurs - New Wave and/or After Murder Park (the first seems to be one of the big canonical albums, and has some very good singles; the second seems to fit in with the "comedown" feel of a lot the later Britpop albums, like Pulp's This Is Hardcore)
Saint Etienne - Foxbase Alpha and/or So Tough (I don't think of them as Britpop, but some people do, and these capture that London '90s hipness that I think was integral to the Britpop scene)
Kenickie - At The Club (Nothing groundbreaking, but it's fun and definitely has that Britpop guitar sound, and would ensure that Elastica aren't your only female representation)
Lightning Seeds - Jollification (I don't know if they're always considered part of Britpop, maybe because they sometimes did music that was pretty. But I think they're close enough. They did a song about being disappointed by England's soccer team, which seems quintessentially Britpop)
Some other acts you could consider:
Shampoo - We Are Shampoo (the Britpop sound reduced to shouty annoying pop, which is the logical conclusion of it - so I'd consider this essential to understanding the genre even though it's not very good)
Cornershop - a reminder that not all British people are white. But sonically I don't know if this really counts, and I wouldn't include them over Saint Etienne.
Manic Street Preachers - I guess it doesn't seem people really consider them part of this scene, but their '90s stuff strikes me as in this wheelhouse.
Charlatans - I see their name a fair amount. I'm guessing they're bad but maybe they're not?
Echobelly - I've only dabbled but they seem decent (and would up your nonmale and nonwhite representation)