There's no correct or incorrect pronunciation. Both of them are correct and sound natural to me. Most dictionaries pronounce it with two syllables, however, Merriam-Webster pronounces it with three syllables.
The pronunciation given by Merriam-Webster is /han-dᵊl-ər/, with three syllables. Here, the second syllable is formed by the syllabic /l/.
We've learnt that a syllable has to have a vowel, but that's not necessary, we have syllabic consonants that form syllables on their own.
Sonorants (/m n l/ and /r/ in AmE etc) often form a syllable on their own whenever they're preceded by obstruents (/s z t d k g p/ etc). Most people tend to insert an extra syllable in words having an obstruent followed by a sonorant. So handler can either be disyllabic or trisyllabic, depending on the speaker. The same goes for battling, rhythmic, handling etc.
The second syllable in rhythm is formed by the syllabic /m/: [ˈɹɪ.ðm̩].
In handler, the obstruent /d/ is followed by the sonorant /l/, so it's trisyllabic for most speakers.
Another similar example would be battling; most people pronounce it with two syllables, but the pronunciation with three syllables is also common: [ˈbæ.tl̩.ɪŋ], here the second syllable is formed by the syllablic l.
The second syllable in handler sounds strong because it's syllabic and syllabic consonants are often longer and more prominent than normal. Here the air is released laterally. Normally, the stops /t/ and /d/ are released by releasing the closure from the centre of the tongue; however, in lateral release, the closure is released from the sides of the tongue.
Button is usually pronounced [ˈbʌ.tnn̩] with a nasal release, here the stop (/t/) isn't released normally but by lowering the velum and allowing air to escape through the nose.
In dictionaries, syllabic consonants are represented by writing a superscript schwa before the consonant: [ˈhæn.dᵊl.ə(ɹ)]. In phonetic transcription, it's represented by a small vertical line below the syllabic consonant: [hæn.dl̩.ə(ɹ)]
