I’ve got a feeling this engages him more.
I’ve a feeling this engages him more.
I got a feeling this engages him more.
To my ear, the first two have the same meaning, but "I have" is more formal" and "I've got" more conversational. I wouldn't personally use "I've" for "I have" except in a construction like "I've been thinking" -- is that the past particilple? -- but using it as this example does might be perfectly common British usage.
To me the third one sounds correct only as past tense -- "As soon as I saw him, I got a feeling of forboding." I wouldn't use "I got" to mean "I have," but only "I received."