How would I say "I can't wait to meet with your little king".

My guess would be:
Ní féidir liom fionraigh le buail le do rí beag.

But as always, I could be wrong and wouldn't mind suggestions for alternative ways to say this.
For context, a friend just had a baby boy named Théoden, who is one of the kings from Lord of the Rings.

You need verbal nouns: to wait = fionraí (or: fanacht, feitheamh), to meet = bualadh

Ní féidir liom fionraí le bualadh le do rí beag.

Oh, should have known to use verbal nouns instead of root forms. Thanks for the reminder.

My first thought was: ar bís

Tá mé ar bís bualadh le do rí beag.

I am excited/I can't wait

I've never seen the ar bis construction before.
Is in possible for you to give a simple explanation for it?

Sorry, I'm rarely able to give grammatical explanations. It's just a phrase we use for excitement, or impatience.
https://www.teanglann.ie/ga/fgb/ar_bís

I see "bís" means a screw, vice, or spiral in another dictionary. So maybe it comes from that. I don't know.
Maybe others will know.

Tá mé ar bís ... bualadh

Do you use it without le, chun or ag in between?

I suppose you could use either one, depending on context.

... ag bualadh - ... meeting...
... le/chun bualadh - .... to meet...

Sometimes I don't know these words as often in speech we leave them out.