I have some difficulty with those posters who see this as a black and white exercise. Apart from the obvious stop if well below V1 scenario, there are just too many causes for the posed situation to pick it at/near V1 unless the ASDR is very much less than the ASDA .. in which case one can consider stopping ...
If MELs are not being carried then I think the situation is not likely from a basic systems viewpoint for the typical aircraft .. however, something out of left field certainly can do mischief. For instance, we had a cargo L188 years ago went dark on rotate due to water or whatever in the electrical bits and pieces from previous flights. If I recall it was a case of external water coming in through the main freight door during a turnaround and the cabin sealing not quite being up to it on the day .. very much a case of sometimes the dog of fate simply lifts its leg ...
If the reasonable assessment (that typical collect data, evaluate, decide in 0.3519 seconds sort of thing) is that the bird isn't going to fly .. then it might be better to takes one's chances in the overrun regardless .. if the bird goes in vertically half a mile off the runway head, the outcome is a foregone conclusion .. all comes back to playing the numbers and rolling the die.
However .. unless the overrun is very attractive and I knew that for a fact .. think I'd opt for taking my chances on continuing ...
This sort of thing probably is going to be in the Sioux City league of incidents/accidents, I guess ..