Deep question. More so than what it appeared to be on first look. I didn't realise English lacks a word for the edible part of a fruit that covers all cases, but it is more contextual than I expected. This means it assumes an understanding of what parts of fruit are edible whenever we are writing or speaking about it - which makes the conventions somewhat non-inclusive.
Scientifically, you are generally describing damage to what is considered to be the Mesocarp of a fruit. Thats meaningless to most since it is not a general-use term, but I will use it going forward to refer to the specific structure.
More broadly you may be referring to the pericarp; though that is used to describe anything between and including the shell of the seed to the skin of the fruit, and is not useful in the case of an avocado where only the mesocarp is eaten.
Since the skin of some fruit can show a bruise, this may be useful in those cases. Even though it is at once more inclusive of structures that show a bruise and more ambiguous (since it includes one structure that will not bruise) than the use of mesocarp. In that case, though, we will refer to what we see on the skin as being the fruit, even when the skin is not edible; "the banana is bruised" or "the apple is bruised". So,if the exocarp is showing a bruise, it is usually safe to assume that the mesocarp is also bruised.
However in the case you used; avocadoes have the problem where the edible portion; the mesocarp, can have a bruise without showing evidence on the inedible skin; the exocarp, which is the cause of your frustration.
Since fruit covers a broad range of structures, even within the same family perhaps fruit flesh may be the best general-purpose phrasing, or just "flesh".
Though that is clearly a semantically overloaded word, the issue can be somewhat corrected by naming the fruit, eg "avocado flesh", though nobody would say "apple flesh" or "strawberry flesh".
This may be because the skin of those other fruits is frequently eaten, whereas only the giant tree sloth would eat the skin of an avocado, and they are sadly extinct and cannot contribute their experience to this question.
Perhaps the mesocarp of a fruit is only considered "flesh" in cases where the skin, or exocarp, is not commonly eaten; eg. "avocado flesh" "watermelon flesh" "orange flesh".
So I would suggest that you could say something like "The avocado flesh turned out to be bruised, but you did not know beforehand because the skin does not show a bruise."
I think this is weak due to convention, and I do not generally hear people describing the mesocarp of those fruits in that way. So it is not common phrasing (at least where I am) and will probably sound a little odd to many people, but they will know what you mean.
By convention, it seems to be more common that in fruits with an inedible rind or skin, the name of the fruit serves double-duty when referring to only the edible portion.
eg. "Hey I wanted to make some smoothie with this avocado (contextually, the entire fruit) but I found it over-ripe and part of the avocado (contextually, the flesh) has changed color." - though in the second usage, you would likely use the word "it".
For additional interest - the structures of a fruit, even within the same family of plants, can be expressions of different parts of the flower after pollination, so this cannot be assumed. - http://fruitandnuteducation.ucdavis.edu/generaltopics/AnatomyPollination/Fruit_Anatomy/
A reference I found only after I had written my reply above may be of interest to you: http://www.ucavo.ucr.edu/General/FruitBerry.html