Why is the syllable division in the word "experience" ex-pe-ri-ence, and not ex-per-i-ence?
Because when Webster first came up with the system of dividing words into syllables, experience was pronounced ex-pee-ri-ence and not ex-peer-i-ence, the way many of us do today. And they haven't bothered changing it since.
Experience: (ĕks-pē′rĭ-ens) — /ɛksˈpiː.rɪ.əns/.
Experiment: (ĕks-pĕr′ĭ-ment) — /ɛksˈpɛr.ɪ.mənt/.
I've translated Webster's notation into IPA, as nobody uses their 1892 phonetic notation anymore.
The American Heritage Dictionary's syllabifications are quite strange. They have:
I would guess they kept the original syllabification of experience for the sake of consistency, but they have updated all the inflections of the word to match the current pronunciation.
Why do you assume ex-pe-ri-ence is wrong? That is how I say it. Therein lies the answer to your question. Because of different dialects and regional variances, there is no one true syllable division. Lexicographers basically choose the pronunciation they believe to be most common. They are usually correct, and their guidance is helpful. But 5 years from the dictionary's release, pronunciation may well have changed.
English is defined from the ground up. There is no governing body to declare standards. English educators, style guides, grammarians, lexicographers, etc., do their best atop constantly shifting sands.