I received a private email on this issue with some quick observations on this matter
A few thoughts -
-R Saadiah has aramaic, be-umra
-Mahzor Vitry p 301 you just say the day, without saying ba or la omer *
-kol bo: laomer
-R. Nissim Gaon laomer
-Rashba: laomer quoted in Beit Yosef
If Beit Yosef would have disagreed with this, he would have said something. So we think R. Yosef Caro would say "la"
But Shulchan Aruch has the word Baomer, but it is in parenthesis **
I looked at early editions on line, same thing. I don't understand why there is parenthesis around it, even in early editions. - I don't understand who puts in parenthesis
-the matzevah (cemetery stone) of the Rama supposedly has la-omer (he died on this day)
a separate issue is to look at early references to Lag Baomer and see if they call it Lag LaOmer or Lag Baomer
* as SA seems to have it without "Omer" - AIUI in context of a B'rachah the word "Omer" is really not needed.
** I explained to him that AFAIK the Ba'Omer in parentheses is a hagaha from the Rema
As I privately emailed a colleague, LA seems to be favored by Rishonim and BA by Acharonim. The above research seems to confirm this.
Shalom and Regards,