A mighty gibbering mouther deals more damage with its bite, but its mightiness likely shouldn't extend to its interior mouths
The damage dealt by a gibbering mouther's special attack engulf remains unchanged by changes to the gibbering mouther's bite attack's damage. They're two independent damage sources. The damage listed for the special attack engulf is the damage dealt by the creature's interior after the creature has successfully used the special attack swallow whole (that the mouther's special attack engulf largely duplicates).
While it may be little consolation to Tibbs who desired his mouther be mighty inside and out, Tibbs's mighty mouther will deal its bite damage to Tibbs's foolish foe twice before it engulfs that foe because that's what's necessary to meet the requirements of swallowing a creature whole. Nonetheless, no matter how mighty Tibbs's mouther, its engulf damage should remain the same as any other mouther's.
That said, a foe that a gibbering mouther successfully engulfs is described as being subject to "several of [the mouther's interior] mouths [that] continue to feed [on] and drain [the foe's] blood." Seeing as how the typical gibbering mouther has 6 exterior bite attacks that each deal 1d4 points of damage and how a typical gibbering mouther's interior deals (coincidentally?) 6d4 points of untyped damage (untyped being important here as most creatures deal acid damage with their swallow whole special attacks), a generous (?) GM could rule that each of those six d4s of engulf damage are, in fact, actually bites, and any modifiers to the gibbering mouther's exterior bites apply to each of those d4 interior bites separately! (This, by the way, may result in a substantial increase to a mouther's engulf damage!)
Such a ruling runs contrary to most other creatures' swallow whole special abilities (see below) and isn't supported explicitly by the gibbering mouther's text beyond that brief mention in the description. Further, most folks who've been engulfed by a gibbering mouther are—let's face it—probably doomed already. However, despite all this, this reader doesn't find this a completely and utterly abhorrent house rule… in the abstract. This player, on the other hand, would be irked were this house rule sprung upon him after his halfling rogue PC were engulfed by Tibbs's mighty pet mouther.
Note: Just to be clear, it seems that creatures typically don't realize any modifiers from their Strength scores on their swallow whole special attacks. For example, the Strength 24 tendriculos's swallow whole damage, the Strength 27 jubear's swallow whole damage, and the Strength 38 hunting horror's swallow whole damage are all unaffected by those creatures' Strength scores. I strongly suspect that the gibbering mouther is no exception to the typical creature with a similar ability.