Not really, as long as there are sufficient females from other, non-genetically altered branches, those strains should be more 'successful' in the quest for survival and eventually the genetically engineered ones will die out,
You are misunderstanding how this works:
The whole POINT of this is that is tips the balance of the scales: you start with countless non-genetically altered mice, and throw in a handful of engineered ones that only breed males, and will pass on the trait to their offspring.
The starting point on the island will be roughly 50% male, 50% female. All of the engineered mice will only create more males, no females. They will mostly breed with random, non-engineered mice, creating more engineered male mice in the process.
Now all of a sudden the the balance of mice on the island a generation later is 51% male, 49% female. Those 1% extra males will also pass on the all-male feature to their offspring as well, increasing the percentage of engineered mice and decreasing the percentage of 'normal' mice. The generation after that may be 53% male, 47% female. A few dozen generations later you will be close to seeing 100% male and 0% female. The chances of any random pairing of mice birthing female offspring becomes vanishingly small.
Existing females die of old age or predation without new females to replace them. Population numbers crash, and the species dies off completely on the island, except for maybe some small, physically isolated groups
Don't forget that each mice can create TONS of offspring, and those all interbreed again. They typically have 5-8 offspring at a time, and can have 5-10 litters a year. This happens FAST. The engineered feature will spread exponentially across the population, with no stopping it. It's an avalanche.