Your first link from the Cambridge Dictionary appears basically correct:
US /ˈkʌm.fɚ.t̬ə.bəl/ UK /ˈkʌm.fə.tə.bəl/
Note what this says, with the funny character "ɚ". Namely, the US version includes an "r" sound. Although not always since there are many regional dialects.
Next, the Merriam Webster dictionary, "America's most trusted online dictionary for English word definitions", proposes:
comfortable /ˈkʌmftɚbəl/ adjective
Which corresponds with the US version from the Cambridge Dictionary. It again includes the character "ɚ", implying an "r" sound.
Actually, that's fascinating... There is a difference between them.
While both dictionaries get there's an "r" sounds, Merriam is showing a more relaxed pronunciation which flips the letters around. "fter" instead of "fort". This depends on whether the word is strongly enunciated, or whether it's spoken quickly.
Then, from wiktionary:
(General American, Canada)
enPR: kŭmf'təbl, kŭmf'tərbl, kŭm'fətəbl, kŭm'fərtəbl
IPA(key): /ˈkʌmf.tə.bəl/, [ˈkʰʌɱ.ftə.bɫ̩], [ˈkʰʌɱ.fɾɚ.bɫ̩]
Which indicates the "r" is by no means guaranteed.
So, I believe that in the most standardized American English, as seen on television news, the "r" is there. In BrE, it's not.
while BrE tends to pronounce full sounds.
Wouldn't this be backwards? In this particular case, BrE is dropping the letter, so not "full" pronunciation of the spelling.
Re-reading your question, it may be you were not concerned about the "r", and were asking about # of syllables.
com-for-ta-ble or comf-ter-ble
I believe this depends on how quickly you are speaking, and whether you are trying to carefully enunciate every syllable. The same person might say it both ways. And, it varies with regional dialect.