Pronunciation of 'mama' in American English with æ
Solution 1:
In the Allman Brothers tune, to my ears the first "mama" starts as one would expect, but Gregg tenses his voice for effect, essentially fronting the vowel. The second "mama" really does sound like cat (TRAP vowel) but likely would be perceived by the audience as the same sort of affect.
This kind of funky jazz/blues always lives, for white performers, somewhere between Coastal South and African American Vernacular. No dialect group in the South, however says "mama" or "momma" with the TRAP vowel, but like PALM, some speakers, both black and white,with a glide on the first syllable. But jazz and blues singers do a lot of bendy things with vowels, so chalk it up to that.
The TRAP-vowel is reserved in the South for one of many pet/babytalk names for grandparents: mamaw and papaw, vowels like "catpaw," but only if COT and CAUGHT aren't pronounced the same for you.
Gregg Allman's natural accent was a Coastal South base with lots of Inland/Mountain South features (think Dolly Partin or Willy Nelson).
As for Funky Mama Moose, the effectiveness of the lyrics depends on an effect called semantic satiation, where something is repeated — in this case an already nonsensical funky momma moose — until it becomes mere sound and rhythm. Pronouncing the first syllable of mama with the TRAP vowel hastens this effect.
Rest assured that neither of these artists called their mothers anything like Mama Moose.