Could you come up with a jazzy chord progression and some lyrics for a love song about a girl that likes frogs just a bit too much?
Sure, I’ll try to create that. Here is a possible chord progression and some lyrics for a love song about a girl that likes frogs just a bit too much:

Verse 1:
Cmaj7  Am7  Dm7  G7
She's got a passion for amphibians
Cmaj7  Am7  Dm7  G7
She loves them more than any other man
Cmaj7  Am7  Dm7  G7
She spends her days in the pond and the swamp
Cmaj7  Am7  Dm7  G7
She kisses every frog hoping one will turn into a prince

Chorus:
Fmaj7  Em7  Am7  Dm7
But I don't need a crown to love her
Fmaj7  Em7  Am7  Dm7
I don't mind her slimy friends
Fmaj7  Em7  Am7  Dm7
She's the only one who makes me hoppy
G7
She's my froggy girl

Verse 2:
Cmaj7  Am7  Dm7  G7
She's got a collection of tadpoles and eggs
Cmaj7  Am7  Dm7  G7
She knows every species by their legs
Cmaj7  Am7  Dm7  G7
She sings them lullabies every night
Cmaj7  Am7  Dm7  G7
She dreams of living in a fairy tale with her frog prince

(Repeat chorus)

Bridge:
Bbmaj7  Abmaj7  Gm7  C9
Sometimes I wonder if she loves me or the frogs more
Bbmaj7  Abmaj7  Gm7  C9
Sometimes I wish I could be more like them
Bbmaj9 Abmaj9 Gm9 C9 
But then she looks at me with those big green eyes 
Bbmaj9 Abmaj9 Gm9 C9 
And she tells me I'm her favorite human

(Repeat chorus)

Outro:
Cmaj9 Am9 Dm9 G13 
She's my froggy girl 
Cmaj9 Am9 Dm9 G13 
She's my froggy girl 
Cmaj9 Am9 Dm9 G13 
She's my froggy girl 
Cmaj9 
Ribbit!
What instrument should I play this on? Like, what would sound the most romantic?
Some romantic instruments you might consider could be the piano, acoustic guitar, tenor saxophone, singing the music in your own voice like a vocalist.

Ultimately, the person or people you are sharing a romantic moment with are likely to be delighted by you using any instrument to share a song with them, as long as you make effort to perform your best and your it comes from your heart.