WDYT of:
[name_f]Lyra[/name_f] [name_f]Bethany[/name_f]
[name_f]Lyra[/name_f] [name_f]Megan[/name_f]
[name_f]Lyra[/name_f] [name_f]Wendy[/name_f]
[name_f]Lyra[/name_f] [name_f]Noemi[/name_f]
[name_f]Lyra[/name_f] [name_f]Idony[/name_f]
[name_f]Lyra[/name_f] [name_f]Josephine[/name_f] - theres at least dozen [name_f]Lyra[/name_f] [name_f]Josephine[/name_f] in the world, seems like popular choice
[name_f]Lyra[/name_f] [name_f]Imogen[/name_f]
Since [name_f]Lyra[/name_f] has prominent r and l sounds, I need middle without l and r sounds, preferably longer.
Think they all flow except for [name_f]Imogen[/name_f], and [name_f]Megan[/name_f] is a bit fast-moving/ abrupt after it. Dare I say [name_f]Josephine[/name_f] sounds the best, but [name_f]Bethany[/name_f] and [name_f]Noemi[/name_f] sound good too. Actually, maybe [name_f]Bethany[/name_f] sounds the best!
[name_f]Lyra[/name_f] is a tricky one in terms of middles. She has a very specific image to me because of the His Dark Materials character that I find difficult to match. I think [name_f]Lyra[/name_f] [name_f]Josephine[/name_f] or [name_f]Lyra[/name_f] [name_f]Imogen[/name_f] would be my favourites. They have the feistiest, slightly adventurous feel out of all of them, so work well with [name_f]Lyra[/name_f].
I know I suggested [name_f]Bethany[/name_f], but I like [name_f]Lyra[/name_f] [name_f]Josephine[/name_f] the most. [name_f]Lyra[/name_f] [name_f]Bethany[/name_f] does have a great flow to it, though! Also partial to [name_f]Lyra[/name_f] [name_f]Noemi[/name_f]. 
From your list, I like:
[name_f]Lyra[/name_f] [name_f]Noemi[/name_f]
[name_f]Lyra[/name_f] [name_f]Idony[/name_f]
[name_f]Lyra[/name_f] [name_f]Josephine[/name_f]
I also like combinations that utilize a D, and I tend to disagree that you need to avoid an L sound.
[name_f]Lyra[/name_f] [name_f]Daphne[/name_f]
[name_f]Lyra[/name_f] [name_f]Adelaide[/name_f]
[name_f]Lyra[/name_f] [name_f]Endellion[/name_f]
[name_f]Lyra[/name_f] [name_f]Alexandrine[/name_f]
[name_f]Lyra[/name_f] [name_f]Desdemona[/name_f] 
[name_f]Lyra[/name_f] [name_f]Bethany[/name_f] is my favourite of your list.
[name_f]Lyra[/name_f] [name_f]Myfanwy[/name_f]?
What about …
[name_f]Lyra[/name_f] [name_f]Anne[/name_f]
[name_f]Lyra[/name_f] [name_f]Bethany[/name_f]
[name_f]Lyra[/name_f] [name_f]Cate[/name_f]
[name_f]Lyra[/name_f] [name_f]Jewel[/name_f]
[name_f]Lyra[/name_f] [name_u]Whitney[/name_u]
[name_f]Lyra[/name_f] [name_f]Imogen[/name_f] is the best of those
Dislike Myfawny. I think I would go with [name_f]Noemi[/name_f].
The only one I can think of with what you want is [name_f]Lyra[/name_f] [name_f]Constance[/name_f] but I like [name_f]Lyra[/name_f] [name_f]Eleanor[/name_f].