Any babbit style bearing will need to have some amount of "endplay". I don't think it is likely a problem. Unless this is a ball bearing turbo. If there were no endplay, it would bind up when hot. Some babbit bearing applications call for as much (or more) as .100" which is nearly 1/8". I am sure they will be able to tell you if it's a problem, but if they tell you it is not, I would believe them.
The measurement to worry about is not end play, but deflection. And you will need a dial indicator to check that. I would be worried if it were more than .003" deflection. That's just off the top of my head, though. Keep in mind that the turbocharger shaft/turbines will be pushed to one side always when operating which will make the shaft endplay disappear completely for practical applications. So the endplay is kind of irrelevant.
The measurement to worry about is not end play, but deflection. And you will need a dial indicator to check that. I would be worried if it were more than .003" deflection. That's just off the top of my head, though. Keep in mind that the turbocharger shaft/turbines will be pushed to one side always when operating which will make the shaft endplay disappear completely for practical applications. So the endplay is kind of irrelevant.